The Ohio State University Number Theory Seminar

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The Ohio State University Number Theory Seminar
Upcoming talk 3/3/2025 - Freddy Saia (University of Illinois Chicago) - Explicit surjectivity for Galois representations of products of elliptic curves over function fields (abstract)
Mondays 4:00pm MW 154 (building photo)

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March 2025

April 2025

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Organizers

Jim Cogdell
Ghaith Hiary
Roman Holowinsky
Wenzhi Luo
Jennifer Park
Stefan Patrikis
Warren Sinnott

Scheduled Talks

Seminars will be a mix of in-person and remote talks, as indicated in the weekly schedule below. For the link, please email patrikis.1 at osu.edu.

3/3/2024 - Freddy Saia (UIC) - Explicit surjectivity for Galois representations of products of elliptic curves over function fields

(abstract)

Abstract: It is natural to study the torsion of an elliptic curve via its Galois representations, which encode the action of Galois on torsion points. Serre’s Open Image Theorem tells us that for an elliptic curve E over a number field K, the mod-

$\ell$ Galois representation is surjective for all sufficiently large primes

$\ell$ . Since Serre’s work, considerable effort has gone towards obtaining extensions to other classes of abelian varieties and also towards making “sufficiently large” effective. I will discuss joint work with Alina Cojocaru, in which we obtain an explicit surjectivity result for products of elliptic curves over function fields. This comes from careful use of techniques of Masser—W{\"u}stholz from the number field setting, in combination with a result of Cojocaru—Hall for elliptic curves over function fields and recent isogeny bounds for elliptic curves over function fields due to Griffon—Pazuki. As an application, we prove that most specializations of certain families of products of elliptic curves over the rationals have no large exceptional primes.

2024-2025

12/2/2024 (Zoom) - Shilpi Mandal (Emory) - Strong u-invariant and Period-Index Bounds (abstract)
For a central simple algebra A over a field K, there are two major invariants, viz., period and index. For a field K, the Brauer-l-dimension of K for a prime number l, is the smallest natural number d such that for every finite field extension L/K and every central simple L-algebra A (of period a power of l), we have that index(A) divides period(A)^d. If K is a number field or a local field, then classical results from class field theory tell us that the Brauer-l-dimension of K is 1. This invariant is expected to grow under a field extension, bounded by the transcendence degree. Some recent works in this area include that of Harbater-Hartmann-Krashen for K a complete discretely valued field, in the good characteristic case. In the bad characteristic case, for such fields K, Parimala-Suresh have given some bounds. Also, the u-invariant of K is the maximal dimension of anisotropic quadratic forms over K. For example, the u-invariant of $\mathbb{C}$ is 1, for F a non-real global or local field the u-invariant of F is 1, 2, 4, or 8, etc. In this talk, I will present similar bounds for the Brauer-l-dimension and the strong u-invariant of a complete non-Archimedean valued field K with residue field $\kappa$ .
11/25/2024 - Asimina Hamakiotes (UConn) - Abelian extensions arising from elliptic curves with complex multiplication (abstract)
Let $K$ be an imaginary quadratic field, and let $\mathcal{O}_{K,f}$ be an order in $K$ of conductor $f \geq 1$ . Let $E$ be an elliptic curve with complex multiplication by $\mathcal{O}_{K,f}$ , such that $E$ is defined by a model over $\mathbb{Q}(j(E))$ , where $j(E)$ is the $j$ -invariant of $E$ . Let $N\geq 2$ be an integer. The extension $\mathbb{Q}(j(E), E[N])/\mathbb{Q}(j(E))$ is usually not abelian; it is only abelian for $N=2,3$ , and $4$ . Let $p$ be a prime and let $n\geq 1$ be an integer. In this talk, we will classify the maximal abelian extension contained in $\mathbb{Q}(E[p^n])/\mathbb{Q}$ .
11/4/2024 - Jacob Mayle (Wake Forest) - Effective Results on Galois Representations of Elliptic Curves (abstract)
Let $E/\mathbb{Q}$ be an elliptic curve without complex multiplication. For a prime $\ell$ , the absolute Galois group acts naturally on the $\ell$ -adic Tate module of $E$ . The $\ell$ -adic Galois representation describes this action and allows us to study aspects of $E$ through the lens of Galois theory. In this talk, we will discuss some applications of the $\ell$ -adic Galois representation and then focus on recent effective results related to it. Specifically, we will discuss a bound (in terms of the conductor of $E$ ) on the largest $\ell$ for which the $\ell$ -adic Galois representation can be non-surjective, as well as a uniform bound on the smallest surjective prime. The former result is joint with Tian Wang and the latter is joint with Tyler Genao and Jeremy Rouse.
10/28/2024 - Benjamin York (UConn) - Models of CM elliptic curves with prescribed $\ell$ -adic Galois image (abstract)
In 2015, Zywina provided Weierstrass models of elliptic curves with complex multiplication (CM) over $\mathbb{Q}$ with a prescribed image of its mod- $\ell$ Galois representation. In 2018, Lozano-Robledo provided a classification for $\ell$ -adic Galois representations attached to elliptic curves with complex multiplication (CM). In this talk, we will discuss a classification of Weierstrass models for CM elliptic curves with specified $\ell$ -adic Galois representation, and discuss our methods for proving this classification. This is joint work with Enrique Gonz\'alez-Jim\'enez and \'Alvaro Lozano-Robledo.
10/21/2024 - HAAR Seminar (Simon Marshall, University of Wisconsin)
10/14/2024 - HAAR Seminar (Brad Rodgers, Queen's University)
10/7/2024 (Zoom) - Deewang Bhamidipati (UC Santa Cruz) - Strata intersections in some unitary Shimura varieties (abstract)
Unitary Shimura varieties are moduli spaces of abelian varieties with a certain extra structure which includes a signature condition. An effective way to understand these spaces is by stratifying them, of which two are of interest: the Ekedah-Oort (EO) stratification, defined with respect to the p-torsion group scheme structure up to isomorphism, and the Newton stratification, defined with respect to the p-divisible group structure up to isogeny. We take a specific stratum in the Newton stratification - the supersingular stratum - and we study its intersection with the EO stratification in some low signature cases.
9/16/2024 - Ryan Chen (MIT) - Fourier coefficients and arithmetic 1-cycles (abstract)
Arithmetic Siegel--Weil formulas predict that derivatives of Eisenstein series Fourier coefficients should encode arithmetic volumes of special cycles on Shimura varieties. Such formulas are arithmetic versions of formulas for complex volumes of moduli spaces in terms of L-functions and Eisenstein series. These "geometrize" the classical Siegel mass and Siegel--Weil formulas, on lattice and lattice vector counting. I will introduce these ideas and overview my proof for arithmetic 1-cycles on unitary Shimura varieties. The key input is a new "limit phenomenon" relating positive characteristic intersection numbers and heights in mixed characteristic.
8/26/2024 - John Yin (OSU) - A Chebotarev density theorem over local fields (abstract)
I will present an analog of the Chebotarev Density Theorem which works over local fields. As an application, I will use it to prove a conjecture of Bhargava, Cremona, Fisher, and Gajović. This is joint work with Asvin G and Yifan Wei.

2023-2024

4/15/2024 (Zoom) - Ross Paterson (University of Bristol) - Quadratic twists as random variables (abstract)
Let E/Q be an elliptic curve and K be a quadratic field of discriminant d. It is well known that the rank of E(Q) and the rank of the quadratic twist E_d(Q) sum to the rank of E(K) — but do these groups generate E(K)? The answer to this question is sometimes yes and sometimes no, which leads to the natural question: how common is each answer? We will provide partial answers, in particular showing that one answer is a lot more likely than the other! In studying this question we are drawn to consider the behaviour of the 2-Selmer groups of E and E_d. There is a heuristic model of Poonen-Rains for the behaviour of these 2-Selmer groups individually, as E varies, but how independent are they? We'll describe a heuristic in this direction, and some results in support of it.
4/8/2024 (9:15am, CH228) - Olivia Beckwith (Tulane University) - Imaginary quadratic fields with p-torsion-free class groups and specified split primes (abstract)
We study Ramanujan-type congruences for Hurwitz class numbers using harmonic Maass forms. As an application, we show that for any odd prime p and finite set of odd primes S, there exists an imaginary quadratic field which splits at each prime in S and has class number indivisible by p. This result is in the spirit of results by Bruinier, Bhargava (when p=3) and Wiles, but the methods are completely different. This is joint work with Martin Raum and Olav Richter.
4/1/2024 - Yujie Xu (Columbia University) - On the geometry of integral models of Shimura varieties (abstract)
Shimura varieties are moduli spaces of abelian varieties with extra structures. Over the decades, various mathematicians have constructed nice integral models of Shimura varieties. In this talk, I will discuss some motivic and deformation-theoretic aspects of integral models of Hodge type (or more generally abelian type). It turns out that their motivic aspects are related to the Hodge Conjecture for abelian varieties. I will talk about my work on improving the construction of these integral models, which have nice applications to abelian varieties and their cycles. If time permits, I will sketch how to deduce a new CM (complex multiplication) lifting result, as well as various other applications to the geometry of Shimura varieties.
3/18/2024 - Abbey Bourdon (Wake Forest University) - An Algorithm for Isolated $j$ -invariants (abstract)
The modular curve $X_1(n)$ is an algebraic curve over $\mathbb{Q}$ whose non-cuspidal points parametrize isomorphism classes of pairs $(E,P)$ where $E$ is an elliptic curve and $P \in E$ is a point of exact order $n$ . We wish to understand isolated points on this curve, which are roughly those not belonging to an infinite family of points of the same degree which are parametrized by a geometric object. For fixed $n$ , there are only finitely many isolated points on $X_1(n)$ of any degree, but as $n$ ranges over all positive integers, there exist infinitely many isolated points (associated to infinitely many non-isomorphic elliptic curves). In this talk, I will discuss a new algorithm which can be used to determine whether an elliptic curve with rational $j$ -invariant gives rise to an isolated point on some modular curve $X_1(n)$ . This builds on prior work of Zywina, which gives a method for computing the image of the adelic Galois representation of a non-CM elliptic curve over $\mathbb{Q}$ . Running the algorithm on all elliptic curves presently in the LMFDB and the Stein-Watkins Database gives evidence for the conjecture that $j\in \mathbb{Q}$ is the image of an isolated point if and only if $j$ corresponds to an elliptic curve with complex multiplication or $j=-140625/8, -9317,$ $351/4$ , or $-162677523113838677$ . This is joint work with Sachi Hashimoto, Timo Keller, Zev Klagsbrun, David Lowry-Duda, Travis Morrison, Filip Najman, and Himanshu Shukla.
3/25/2024 (HAAR seminar)- Salim Taylou (Harvard) - Mixed mock modularity of special divisors
2/26/2024 - John Voight (Dartmouth) - On abelian varieties whose torsion is not self-dual (abstract)
We construct infinitely many abelian surfaces defined over the rational numbers whose ell-torsion is not isomorphic as a Galois module to that of the dual for ell = 2,3,5,7. This is joint work with Sarah Frei and Katrina Honigs.
1/22/2024 (HAAR seminar)- Mike Daas (Leiden) - A p-adic analogue of a formula by Gross and Zagier (Zoom)
1/8/2024 (HAAR seminar)- Jerry Wang (Waterloo) - On the squarefree values of $a^4 + b^3$ (Zoom)
11/27/2023 - Julian Lyczak (University of Bath) - Paucity of rational points on fibrations with multiple fibres (Zoom) (abstract)
This talk is based on joint work with Browning and Smeets and concerns the density of everywhere locally soluble members within a family of varieties. Our work treats the most general case, where components of higher multiplicity are allowed. Using exact conditions for a fibre to be locally soluble, coming from logarithmic geometry, we can translate the counting problem into a purely analytical question. Then using the large sieve we obtain upper bounds for the relevant counting problems for several families of examples. In some cases we can also provide lower bounds of the same order, and even asymptotics. Using these results I will discuss different ways in which the Loughran-Smeets conjecture depends on the absence of multiple fibres, and indicate the geometry involved in stating a conjecture for the general setting.
11/20/2023 - Nick Rome (TU Graz) - The arithmetic of the norm one torus (Zoom) (abstract)
Given a finite extension of number fields K/Q, the associated field norm can be used to define an affine variety given by the equation Norm_{K/Q}(x)=1. We will discuss the qualitative distribution of rational points on this variety, and how this distributions changes as one varies the field K.
10/8/2023 - Shiva Chidambaram (MIT) - Computing Q-isogeny classes of principally polarized abelian surfaces. (abstract)
Mazur's isogeny theorem gives an explicit finite set of primes that can occur as the degree of any irreducible rational isogeny of any elliptic curve over Q. Building on this powerful result, there is a complete classification of all possible isogeny classes/graphs of elliptic curves over Q. In this talk, we will investigate isogeny classes in dimension 2. Given a principally polarized abelian surface (PPAS) over Q with no extra endomorphisms, we will describe a practical algorithm to compute all other rational PPASs in its isogeny class. While there is no genus 2 analog of Mazur's theorem, one can still compute exceptional primes for torsion representations of any given PPAS, which is the starting point of our algorithm. I will mention how this can be extended to some other settings like Picard curves as well. Finally, I will discuss the results of running our algorithm on the database of genus 2 curves in the LMFDB, and show the isogeny graphs we obtain. This is joint work with Raymond van Bommel, Edgar Costa and Jean Kieffer.
9/25/2023 - Tyler Genao (OSU) - Uniform bounds on torsion of elliptic curves in geometric isogeny classes (abstract)
A celebrated result of Merel (1996) proved there exists an absolute bound on all torsion groups of elliptic curves over number fields of a fixed degree. On the other hand, any geometric isogeny class of elliptic curves (``geometric" meaning defined over a fixed algebraic closure of $\mathbb{Q}$ ) will contain $j$ -invariants of arbitrarily large degree over $\mathbb{Q}$ . It may be surprising, then, that torsion groups from geometric isogeny classes are still ``uniformly behaved" in several ways. In this talk, we will discuss new results on polynomial bounds for torsion from geometric isogeny classes of elliptic curves, as well as ``typical boundedness" of torsion from the collection of geometric isogeny classes with at least one rational $j$ -invariant.
9/11/2023 - Ariel Weiss (OSU) - Rank growth of elliptic curves over N-th root extensions (abstract)
Fix an elliptic curve $E/\mathbb{Q}$ . If $\mathcal{K}$ is some family of finite extensions of $\mathbb{Q}$ , then, for each field $K\in\mathcal{K}$ , $E(K)$ is a finitely generated abelian group, and it is natural to ask how the rank of $E(K)$ varies as $K$ runs through $\mathcal{K}$ . For example, if $\mathcal{K}$ is the set of all quadratic extensions of $\mathbb{Q}$ , then a conjecture of Goldfeld states that $\mathrm{rk}\ E(K) = \mathrm{rk}\ E(\mathbb{Q})$ for $50\%$ of $K$ , and $\mathrm{rk}\ E(K) = \mathrm{rk}\ E(\mathbb{Q})+1$ for the other $50\%$ . In this talk, we will study the growth of the rank of $E$ after base change to the fields $K_d=\mathbb{Q}(\sqrt[N]d)$ , where $N=2\cdot3^m$ for some fixed $m\ge 1$ . Our main result is that, if $E$ admits a $3$ -isogeny, then the average "new rank" of $E(K_d)$ is bounded as $|d|\to\infty$ .
8/28/2023 - Alex Cowan (Harvard) - A twisted additive divisor problem (abstract)
Shifted convolutions of sum-of-divisors functions measure correlation between divisors of integers with a fixed difference. In this talk, I'll give asymptotics for a shifted convolution of sum-of-divisors functions twisted by Dirichlet characters and with nonzero powers. The spectral methods of automorphic forms used to prove the result are quite general, and I'll present a conceptual overview. One step of the calculation uses a less well-known technique called "automorphic regularization" for obtaining the spectral decomposition of a combination of Eisenstein series which is not obviously square-integrable.

2022-2023

4/17/2023-4/19/2023 - Zassenhaus Lectures Bhargav Bhatt (IAS/Princeton) - Prismatic cohomology and applications
4/13/2023-4/14/2023 - OSU Number Theory Seminar Series
4/10/2023 - Tian Wang (University of Illinois at Chicago) - Effective open image theorem for a product of GSp-type abelian varieties (abstract)
In the 1970s, Serre and Ribet proved the Open Image Theorem (OIT) for a product of non-CM, non-isogenous elliptic curves, asserting that for sufficiently large prime $\ell$ , the mod- $\ell$ Galois representation is surjective in an appropriate matrix group. Since then, a lot of work has been done on the explicit bounds of the largest prime $\ell$ for which the mod- $\ell$ Galois representation is nonsurjective. A similar version of the OIT holds for a product of GSp-type abelian varieties due to the work of Serre and Hindry-Ratazzi. We will give an explicit bound of the largest nonsurjective prime $\ell$ for a product of GSp-type abelian varieties in terms of their standard invariants. This is joint work with Jacob Mayle.
4/3/2023 - HAAR seminar (Nina Zubrilina)
3/27/2023 - HAAR seminar (Maksym Radziwill)
3/20/2023 - Jeremy Booher (University of Florida) - Lifting $G$ -Valued Galois Representations when $\ell \neq p$ (Zoom) (abstract)
We study the universal lifting spaces of local Galois representations valued in reductive group schemes when $\ell \neq p$ . In particular, for a representation of the Galois group of an $\ell$ -adic field valued in the characteristic- $p$ points of a reductive group, we construct a canonical smooth component in the space of lifts under certain technical conditions applicable to any root datum. This generalizes the minimally ramified deformation condition previously studied for classical group and proves the existence of lifts to characteristic zero. Carrying this out requires answering subtle questions about reductive group schemes over DVRs. This is joint work with Sean Cotner and Shiang Tang.
3/6/2023 - Ashwin Iyengar (Johns Hopkins) - On local Galois deformation rings when $\ell=p$ (Zoom) (abstract)
I will speak about joint work with Vytautas Paškūnas and Gebhard Böckle which describes the space of lifts (to characteristic 0, for instance) of a representation (with mod p coefficients) of the absolute Galois group of a finite extension of the p-adic numbers. In the talk, I will try to introduce and motivate the subject of deformation theory in the context of the Langlands program. Then I will state our results and try to illustrate how our proof works using a simple example.
2/27/2023 - Brandon Alberts (Eastern Michigan University) - A Random Group with Local Data (in-person) (abstract)
The Cohen--Lenstra heuristics describe the distribution of $\ell$ -torsion in class groups of quadratic fields as corresponding to the distribution of certain random p-adic matrices. These ideas have been extended to using random groups to predict the distributions of more general unramified extensions in families of number fields (see work by Boston--Bush--Hajir, Liu--Wood, Liu--Wood--Zureick-Brown). Via the Galois correspondence, the distribution of unramified extensions is a specific example of counting number fields with prescribed ramification and bounded discriminant. As of yet, no constructions of random groups have been given in the literature to predict the answers to famous number field counting conjectures such as Malle's conjecture. We construct a "random group with local data" bridging this gap, and use it to describe new heuristic justifications for number field counting questions.
2/20/2023 - Ari Shnidman (Hebrew University) - Bielliptic Picard curves (in-person) (abstract)
I'll describe the geometry and arithmetic of the curves $y^3 = x^4 + ax^2 + b$ . The Jacobians of these curves factor as a product of an elliptic curve and an abelian surface $A$ . The latter is an example of a "false elliptic curve", i.e. an abelian surface with quaternionic multiplication. I'll explain how to see this from the geometry of the curve, and then I'll give some results on the Mordell-Weil groups $A(\mathbf{Q})$ . This is based on joint work with Laga and Laga-Schembri-Voight.
2/13/2023 - Pol van Hoften (Stanford) - Hecke orbits on Shimura varieties of Hodge type (abstract)
Oort conjectured in 1995 that isogeny classes in the moduli space A_g of principally polarised abelian varieties in characteristic p are Zariski dense in the Newton strata containing them. There is a straightforward generalisation of this conjecture to the special fibres of Shimura varieties of Hodge type, and in this talk, I will present a proof of this conjecture. I will mostly focus on the case of A_g since many of the new ideas can already be explained in this special case. This is joint work with Marco D'Addezio.
11/28/2022 - Juanita Duque Rosero (Dartmouth) - Quadratic Chabauty: geometric and explicit (in-person) (abstract)
Geometric quadratic Chabauty is a method, pioneered by Edixhoven and Lido ['21], whose goal is to determine the rational points on a nice curve X. The main tools that this method uses are p-adic analysis and Gm-torsors over the Jacobian of X. In this talk, I will give an overview of the method, focusing on explicit computations. I will also present a comparison theorem to the (original) method for quadratic Chabauty. Finally, we will look at a specific example of a modular curve where the method of geometric quadratic Chabauty can be used. This is joint work with Sachi Hashimoto and Pim Spelier.
11/21/2022 - Christopher Keyes (Emory) - Local solubility in families of superelliptic curves (in-person) (abstract)
If we choose at random an integral binary form $f(x, z)$ of fixed degree $d$ , what is the probability that the superelliptic curve with equation $C \colon y^m = f(x, z)$ has a $p$ -adic point, or better, points everywhere locally? In joint work with Lea Beneish (\url{https://arxiv.org/abs/2111.04697}), we show that the proportion of forms $f(x, z)$ for which $C$ is everywhere locally soluble is positive, given by a product of local densities. By studying these local densities, we produce bounds which are suitable enough to pass to the large $d$ limit. In the specific case of curves of the form $y^3 = f(x, z)$ for a binary form of degree 6, we determine the probability of everywhere local solubility to be 96.94\%, with the exact value given by an explicit infinite product of rational function expressions.
10/17/2022 - Naomi Sweeting (Harvard) - Kolyvagin's Conjecture and Higher Congruences of Modular Forms (in-person) (abstract)
Given an elliptic curve E, Kolyvagin used CM points on modular curves to construct a system of classes valued in the Galois cohomology of the torsion points of E. Under the conjecture that not all of these classes vanish, he deduced remarkable consequences for the Selmer rank of E. For example, his results, combined with work of Gross-Zagier, implied that a curve with analytic rank one also has algebraic rank one; a partial converse follows from his conjecture. In this talk, I will report on work proving several new cases of Kolyvagin's conjecture. The methods follow in the footsteps of Wei Zhang, who used congruences between modular forms to prove Kolyvagin's conjecture under some technical hypotheses. By considering congruences modulo higher powers of p, we remove many of those hypotheses. The talk will provide an introduction to Kolyvagin's conjecture and its applications, explain an analog of the conjecture in an opposite parity regime, and give an overview of the proofs, including the difficulties associated with higher congruences of modular forms and how they can be overcome via deformation theory.
10/10/2022 - Mohamed Tawfik (King's College London) - Brauer-Manin obstruction on Kummer Surfaces (Zoom) (abstract)
We discuss Brauer-Manin obstructions on Kummer surfaces of products of certain CM elliptic curves. We start by putting necessary and sufficient conditions on these surfaces to get a non-trivial transcendental Brauer group, then we find the construction of these groups. Further, using a theorem by Harpaz and Skorobogatov, we show that a non-trivial element of order 5 of the transcendental Brauer group always gives rise to Brauer-Manin obstruction to weak approximation on these surfaces. Finally, we show that for most cases there is no obstruction coming from the algebraic part.
9/26/2022 - Kalyani Kansal (Johns Hopkins) - Intersections of irreducible components of the Emerton-Gee stack for GL_2 (Zoom) (abstract)
Let K be a finite extension of Q_p. In a 2020 paper, Caraiani, Emerton, Gee, and, Savitt constructed a moduli stack of two dimensional mod p representations of the Galois group of K. We compute criteria for codimension one intersections of the irreducible components of this stack, and interpret them in sheaf-theoretic terms.
9/19/2022 - Luochen Zhao (Johns Hopkins) - Explicit period formulas for totally real p-adic L-functions, à la Cassou-Noguès (Zoom) (abstract)
The p-adic Hecke L-functions over totally real fields are known to exist by works of Deligne-Ribet, Cassou-Noguès and Barsky in the late 70s, albeit the whole picture of which is still clouded to this day. In this talk I will report my recent work on the explict determination of the incarnate p-adic measures that generalizes the p-adic Bernoulli distributions, and its applications in the Gross-Stark conjecture and totally real Iwasawa invariants.

2021-2022

4/25/2022 - Shantanu Agarwal (University of Iowa) - A converse theorem without root numbers (abstract)
Broadly, a converse theorem characterizes automorphic forms in terms of analytic properties of associated L-functions. Converse theorems have historically played a key role in proving cases of modularity, and also to establish various instances of Langlands functoriality. Weil proved a converse theorem for modular forms for congruence subgroups of SL(2, Z). In his work, he requires functional equations for L-functions twisted by a family of Dirichlet characters. In Weil's hypothesis, these functional equations must have precise values for the so called root numbers. A recent work of Booker relaxes this condition by allowing arbitrary root numbers in the functional equations. I extend Booker's result by proving an analogous theorem for a rational function field.
4/21/2022 (4:15pm-5:15pm, MW 154 or join by Zoom. Note unusual day!)- Andrew Kobin (Emory University) - Categorifying quadratic zeta functions (abstract)
Zeta functions show up everywhere in math these days. While some work in the past has brought homotopical methods into the theory of zeta functions, there is in fact a lesser-known zeta function that is native to homotopy theory. Namely, every suitably finite decomposition space (aka 2-Segal space) admits an abstract zeta function as an element of its incidence algebra. In this talk, I will show how many 'classical' zeta functions from number theory and algebraic geometry can be realized in this homotopical framework, and briefly advertise an upcoming preprint (joint with Jon Aycock) that categorifies the Dedekind zeta function of a quadratic number field.
3/28/2022 - Aleksander Horawa (University of Michigan) - Motivic action on coherent cohomology of Hilbert modular varieties (abstract)
A surprising property of the cohomology of locally symmetric spaces is that Hecke operators can act on multiple cohomological degrees with the same eigenvalues. We will discuss this phenomenon for the coherent cohomology of line bundles on modular curves and, more generally, Hilbert modular varieties. We propose an arithmetic explanation: a hidden degree-shifting action of a certain motivic cohomology group (the Stark unit group). This extends the conjectures of Venkatesh, Prasanna, and Harris to Hilbert modular varieties.
1/24/2022 - Yunqing Tang (Princeton University) - The unbounded denominators conjecture (abstract)
The unbounded denominators conjecture, first raised by Atkin and Swinnerton-Dyer, asserts that a modular form for a finite index subgroup of SL_2(Z) whose Fourier coefficients have bounded denominators must be a modular form for some congruence subgroup. In this talk, we will give a sketch of the proof of this conjecture based on a new arithmetic algebraization theorem. This is joint work with Frank Calegari and Vesselin Dimitrov.
12/13/2021 - Eun Hye Lee (Stony Brook University) - Subconvexity of Shintani Zeta Functions (abstract)
In this talk, I will introduce the Shintani zeta function and the problem of subconvexity. And then, I will survey the recent results of myself and R. Hough.
12/6/2021 - Lue Pan (Princeton University) - p-adic de Rham Galois representations in the completed cohomology of modular curves (abstract)
In this talk, I want to explain how to use the geometry of modular curves at infinite level and Hodge-Tate period map to study p-adic de Rham Galois representations appearing in the p-adically completed cohomology of modular curves. As an application, we will show that these representations up to twists come from modular forms, which previously was known by totally different methods.
11/29/2021 - Levent Alpöge (Harvard University) - Effective height bounds for odd-degree totally real points on some curves (abstract)
I will give a finite-time algorithm that, on input (g,K,S) with g > 0, K a totally real number field of odd degree, and S a finite set of places of K, outputs the finitely many g-dimensional abelian varieties A/K which are of GL_2-type over K and have good reduction outside S. The point of this is to effectively compute the S-integral K-points on a Hilbert modular variety, and the point of that is to be able to compute all K-rational points on complete curves inside such varieties. This gives effective height bounds for rational points on infinitely many curves and (for each curve) over infinitely many number fields. For example one gets effective height points for odd-degree totally real points on x^6 + 4y^3 = 1, by using the hypergeometric family associated to the arithmetic triangle group \Delta(3,6,6).
11/22/2021 *in-person: MW154*- Sarah Arpin (University of Colorado, Boulder) - Adding Level Structure to Supersingular Elliptic Curve Isogeny Graphs (abstract)
Supersingular elliptic curves have seen a resurgence in the past decade with new post-quantum cryptographic applications. In this talk, we will discover why and how these curves are used in new cryptographic protocol. Supersingular elliptic curve isogeny graphs can be endowed with additional level structure. We will look at the level structure graphs and the corresponding picture in a quaternion algebra.
11/15/2021 - Jacob Tsimerman (University of Toronto) - Unlikely intersections and the Andre-Oort conjecture (abstract)
The Andre-Oort conjecture concerns special points of a Shimura variety S - points that are in a certain sense "maximally symmetric". It states that if a variety V in S contains a zariski-dense set of such points, then V must itself be a Shimura variety. It is an example of the field now known as "unlikely intersections theory" which seeks to explain "arithmetic coincidences" using geometry. In fact, there is a very natural sense in which the Andre-Oort conjecture can be seen as an analogue of Faltings theorem concerning rational points on curves. The proof of this conjecture involves a wide range of disparate mathematical ideas - functional transcendence, mondromy, point counting in transcendental sets, upper bounds for arithmetic complexity (heights of special points), and p-adic hodge theory. We will survey these concepts and how they relate to each other in the proof, aiming to give an overview of the relevant ideas. We will also discuss the current status of the field, now spearheaded by the (still extremely open!) Zilber-Pink conjecture, and what is required to make further progress.
11/8/2021 - No seminar: Danny Krashen (University of Pennsylvania) will speak in the Algebra Seminar.
11/1/2021 - Félix Baril Boudreau (Western University) - Computing an L-function modulo a prime (abstract)
Let $E$ be an elliptic curve with non-constant $j$ -invariant over a function field $K$ with constant field of size an odd prime power $q$ . Its $L$ -function $L(T,E/K)$ belongs to $1 + T\mathbb{Z}[T]$ . Inspired by the algorithms of Schoof and Pila for computing zeta functions of curves over finite fields, we propose an approach to compute $L(T,E/K)$ . The idea is to compute, for sufficiently many primes $\ell$ invertible in $K$ , the reduction $L(T,E/K) \bmod{\ell}$ . The $L$ -function is then recovered via the Chinese remainder theorem. When $E(K)$ has a subgroup of order $N \geq 2$ coprime with $q$ , Chris Hall showed how to explicitly calculate $L(T,E/K) \bmod{N}$ . We present novel theorems going beyond Hall's.
10/26/2021, 10:15am-11:15am (note unusual day and time) - Jack Thorne (University of Cambridge) - Symmetric power functoriality for holomorphic modular forms (abstract)
Symmetric power functoriality is the first interesting case of Langlands’s functoriality conjectures and is closely connected to other problems in number theory, such as the Ramanujan conjecture and the Sato—Tate conjecture for elliptic curves. I will discuss the historical motivation for the problem and joint work with James Newton that solves the problem for holomorphic elliptic modular forms.
10/18/2021 *in-person: MW154*- Pan Yan (OSU) - $L$ -function for $\text{Sp}(4)\times \text{GL}(2)$ via a non-unique model (abstract)
We prove a conjecture of Ginzburg and Soudry (2020 IMRN) on an integral representation for the tensor product partial $L$ -function for $\text{Sp}(4)\times \text{GL}(2)$ , which is derived from the twisted doubling method of Cai, Friedberg, Ginzburg, and Kaplan. We show that the integral unfolds to a non-unique model and analyze it using the New Way method of Piatetski-Shapiro and Rallis.
10/11/2021 - Debanjana Kundu (University of British Columbia) - Arithmetic Statistics of Iwasawa Invariants (abstract)
We will discuss the average behaviour of the Iwasawa invariants for the Selmer groups of elliptic curves over the cyclotomic $\mathbb{Z}_p$ extension of $\mathbb{Q}$ , setting out new directions in arithmetic statistics and Iwasawa theory. This is joint work with Anwesh Ray. If time permits, we will discuss the case of anticyclotomic $\mathbb{Z}_p$ -extensions as well.
10/4/2021 - Ariel Weiss (Hebrew University of Jerusalem) - Prime torsion in the Tate-Shafarevich groups of abelian varieties over $\mathbb{Q}$ (abstract)
Very little is known about the Tate-Shafarevich groups of abelian varieties. On the one hand, the BSD conjecture predicts that they are finite. On the other hand, heuristics suggest that, for each prime $p$ , a positive proportion of elliptic curves $E/\mathbb{Q}$ have $\Sha(E)[p] \ne 0$ , and one expects something similar for higher dimensional abelian varieties as well. Yet, despite these expectations, it seems to be an open question whether, for each prime $p$ , there exists even a single elliptic curve over $\mathbb{Q}$ with $\Sha(E)[p] \ne 0$ . In this talk, I will show that, for each prime $p$ , there exists a geometrically simple abelian variety $A/\mathbb{Q}$ with $\Sha(A)[p]\ne 0$ . Our examples arise from modular forms with Eisenstein congruences. This is joint work with Ari Shnidman.
9/20/2021 - Jacob Mayle (University of Illinois at Chicago) - On the effective open image theorem (abstract)
A celebrated theorem of Serre gives that if $\ell$ is sufficiently large, then the mod $\ell$ Galois representation of a non-CM elliptic curve $E/\mathbb{Q}$ is surjective. Serre asked if the largest non-surjective prime is always at most 37. We'll give an overview of the progress that has been made toward this question and discuss recent progress (joint with Tian Wang) in bounding the largest non-surjective prime.
9/13/2021 - Chi-Yun Hsu (UCLA) - Partial classicality of Hilbert modular forms (abstract)
Overconvergent Hilbert modular forms are defined over a strict neighborhood of the ordinary locus of the Hilbert modular variety. The philosophy of classicality theorems is that when the p-adic valuation of Up-eigenvalue is small compared to the weight (called a small slope condition), an overconvergent Up eigenform is automatically classical, namely it can be extended to the whole Hilbert modular variety. On the other hand, we can define partially classical forms as forms defined over a strict neighborhood of “partially ordinary locus”. We show that under a weaker small slope condition, an overconvergent form is automatically partially classical, employing Kassaei’s method of analytic continuation.

2020-2021

4/19/2021 - Sug Woo Shin (UC Berkeley) - On GSpin(2n)-valued automorphic Galois representations (abstract)
I will present my joint work with Arno Kret, where we construct a GSpin(2n)-valued ell-adic Galois representation attached to a cuspidal cohomological automorphic representation pi of a suitable quasi-split form of GSO(2n) over a totally real field, under the hypothesis that pi has a Steinberg component at a finite place. This uses input from the cohomology of certain Shimura varieties for GSO(2n); as such we need to take a suitable form of GSO(2n) depending on the parity of n. (We take the split form if and only if n is even.)
4/5/2021 - Jiuya Wang (Duke University) - Pointwise Bound for $\ell$ -torsion of Class Groups (abstract)
$\ell$ -torsion conjecture states that $\ell$ -torsion of the class group $|\text{Cl}_K[\ell]|$ for every number field $K$ is bounded by $\text{Disc}(K)^{\epsilon}$ . It follows from a classical result of Brauer-Siegel, or even earlier result of Minkowski that the class number $|\text{Cl}_K|$ of a number field $K$ are always bounded by $\text{Disc}(K)^{1/2+\epsilon}$ , therefore we obtain a trivial bound $\text{Disc}(K)^{1/2+\epsilon}$ on $|\text{Cl}_K[\ell]|$ . We will talk about results on this conjecture, and recent works on breaking the trivial bound for $\ell$ -torsion of class groups in some cases based on the work of Ellenberg-Venkatesh.
3/22/2021 - Hyuk Jun Kweon (MIT) - Bounds on the Torsion Subgroups of Néron-Severi Group Schemes (abstract)
Let $X \hookrightarrow \mathbb{P}^r$ be a smooth projective variety defined by homogeneous polynomials of degree $\leq d$ over an algebraically closed field $k$ . Let $\mathbf{Pic}\, X$ be the Picard scheme of $X$ , and $\mathbf{Pic}^0 X$ be the identity component of $\mathbf{Pic}\, X$ . The N\'eron--Severi group scheme of $X$ is defined by $\mathbf{NS} X = (\mathbf{Pic}\, X)/(\mathbf{Pic}^0 X)_{\mathrm{red}}$ , and the N\'eron--Severi group of $X$ is defined by $\mathrm{NS}\, X = (\mathbf{NS} X)(k)$ . We give an explicit upper bound on the order of the finite group $(\mathrm{NS}\, X)_{{\mathrm{tor}}}$ and the finite group scheme $(\mathbf{NS} X)_{{\mathrm{tor}}}$ in terms of $d$ and $r$ . As a corollary, we give an upper bound on the order of the torsion subgroup of second cohomology groups of $X$ and the finite group $\pi^1_{\mathrm{et}}(X,x_0)^{\mathrm{ab}}_{\mathrm{tor}}$ . We also show that $(\mathrm{NS}\, X)_{\mathrm{tor}}$ is generated by $(\deg X -1)(\deg X - 2)$ elements.
3/1/2021 - Lynnelle Ye (Stanford University) - Properness of Hilbert modular eigenvarieties (abstract)
Can a family of finite-slope modular Hecke eigenforms lying over a punctured disc in weight space always be extended over the puncture? This was first asked by Coleman and Mazur in 1998 and settled (in the affirmative) by Diao and Liu in 2014 using deep, powerful Galois-theoretic machinery. We will discuss a generalization of this result to Hilbert modular eigenvarieties for totally split primes. We do not use Diao-Liu's method. Instead we adapt an earlier method of Buzzard and Calegari based on elementary properties of overconvergent modular forms, building on recent work of Ren-Zhao for the boundary of weight space and Hattori for algebraic weights.
2/22/2021 - Soumya Sankar (OSU) - Counting elliptic curves with a rational N-isogeny (abstract)
The classical problem of counting elliptic curves with a rational N-isogeny can be phrased in terms of counting rational points on certain moduli stacks of elliptic curves. Counting points on stacks poses various challenges, and I will discuss these along with a few ways to overcome them. I will also talk about the theory of heights on stacks developed in recent work of Ellenberg, Satriano and Zureick-Brown and use it to count elliptic curves with an N-isogeny for certain N. The talk assumes no prior knowledge of stacks and is based on joint work with Brandon Boggess.
2/15/2021 - Manami Roy (Fordham University) - Counting cuspidal automorphic representation of GSp(4) and its application. (abstract)
There is a well-known connection between the Siegel modular forms of degree 2 and the automorphic representations of GSp(4). Using this relationship and the available dimension formulas for the spaces of Siegel cusp forms of degree 2, we count a specific set of cuspidal automorphic representations of GSp(4). Consequently, we obtain an equidistribution result for a family of cuspidal automorphic representations of GSp(4). This kind of equidistribution result is analogous to the so-called vertical Sato-Tate conjecture for GL(2). This method of counting automorphic representations is also helpful for computing dimensions of some spaces of Siegel cusp forms, which are not yet known. The talk is based on a joint work with Ralf Schmidt and Shaoyun Yi.
2/1/2021 - Eric Stubley (University of Chicago) - Locally Split Galois Representations and Hilbert Modular Forms of Partial Weight One (abstract)
The Galois representation attached to a p-ordinary eigenform is upper triangular when restricted to a decomposition group at p. A natural question to ask is under what conditions this upper triangular decomposition splits as a direct sum. Ghate and Vatsal have shown that for Galois representations coming from families of p-ordinary eigenforms, the restriction to a decomposition group at p is split if and only if the family has complex multiplication. In their proof, the weight one members of the family play a key role.
I'll speak about work which aims to answer similar questions in the case of ordinary Galois representations for a totally real field which are split at only some of the primes above p. In this work Hilbert modular forms of partial weight one play a central role. I'll discuss what is known about partial weight one forms and the new techniques used in generalizing Ghate and Vatsal's result to this situation.
1/25/2021 - Patrick Allen (McGill University) - Modularity of some PGL(2,5) representations (abstract)
Serre's conjecture, proved by Khare and Wintenberger, states that every odd two dimensional mod p representation of the absolute Galois group of the rationals comes from a modular form. This admits a natural generalization to totally real fields, but even the real quadratic case seems completely out of reach. I'll discuss some of the difficulties one encounters and then discuss some new cases that can be proved when p = 5. This is joint work with Chandrashekhar Khare and Jack Thorne.
11/23/2020 - Caleb Springer (Penn State University) - Abelian varieties and their endomorphism rings (abstract)
The endomorphism ring of an elliptic curve defined over a finite field is an important and widely-studied object which is useful in many contexts, including cryptography, the investigation of isogeny graphs, and the determination of the structure of the group of rational points. In this talk, I will outline some generalizations of these results to abelian varieties of higher dimension, and present a subexponential algorithm for computing the endomorphism ring of an ordinary abelian surface defined over a finite field under certain technical assumptions. The heart of the algorithm, generalizing the method of Bisson and Sutherland for elliptic curves, exploits the ideal class groups of orders in CM fields, with an application of class field theory.
11/9/2020 - Guillermo Mantilla-Soler (Aalto University) - A complete invariant for real S_n number fields (abstract)

In this talk we will review some of the most common invariants from algebraic number theory. We divide number fields in families, by Galois groups or signature, and study the trace form in each collection of fields. We will show how this division on number fields led us to the discovery that for real S_n number fields, with restricted ramification, the trace is a complete invariant.
11/2/2020 - Matilde Lalín (Université de Montréal) - Nonvanishing for cubic L-functions over function fields. (abstract)

Chowla's conjecture predicts that $L (1/2, \chi)$ does not vanish for Dirichlet $L$ -functions associated with primitive characters $\chi$ . It was first conjectured for the case of $\chi$ quadratic. For that case, Soundararajan proved that at least 87.5\% of the values $L (1/2, \chi)$ do not vanish, by calculating the first mollified moments. For cubic characters, the first moment has been calculated by Baier and Young (on $\mathbb{Q}$ ), by Luo (for a restricted family on $\mathbb{Q} (\sqrt{-3})$ ), and on function fields by David, Florea, and Lal\'in. In this talk we prove that there is a positive proportion of cubic Dirichlet characters for which the corresponding $L$ -function at the central value does not vanish. We arrive at this result by computing the first mollified moment using techniques that we previously developed in our work on the first moment of cubic $L$ -functions, and by obtaining a sharp upper bound for the second mollified moment, building on work of Lester and Radziwi\l\l, Harper, and Radziwi\l\l - Soundararajan. Our results are on function fields, but with additional work they could be extended to number fields, assuming GRH. This is joint work with Chantal David and Alexandra Florea.
10/26/2020 - Naser Talebizadeh Sardari (MPIM) - Vanishing Fourier Coefficients of Hecke Eigenforms (abstract)

We prove that, for fixed level (N,p) = 1 and prime p > 2, there are only finitely many Hecke eigenforms f of level \Gamma_1(N) and even weight with a_p(f) = 0 (p-th Fourier coefficient) which are not CM. This is joint work with Frank Calegari.
10/19/2020 - Efthymios Sofos (University of Glasgow) - Schinzel Hypothesis with probability 1 and rational points (abstract)

Joint work with Alexei Skorobogatov, preprint: https://arxiv.org/abs/2005.02998. Schinzel's Hypothesis states that every integer polynomial satisfying certain congruence conditions represents infinitely many primes. It is one of the main problems in analytic number theory but is completely open, except for polynomials of degree 1. We describe our recent proof of the Hypothesis for 100% of polynomials (ordered by size of coefficients). We use this to prove that, with positive probability, Brauer--Manin controls the Hasse principle for Châtelet surfaces.
10/12/2020 - Shiva Chidambaram (University of Chicago) - Abelian surfaces with fixed 3-torsion (abstract)

The Siegel modular variety A_2(3) which parametrizes abelian surfaces with full level 3 structure is birational to the Burkhardt quartic threefold. This was shown to be rational over Q by Bruin and Nasserden. What can we say about its twist A_2(\rho) for a Galois representation \rho valued in GSp(4, F_3)? It is geometrically rational and known to be unirational over Q by a map of degree at most 6. But it is not rational in general. The degree 6 cover corresponds to a choice of an odd theta characteristic. An explicit description of the universal genus 2 curve over this cover is obtained using invariant theoretic ideas. An application of this result is to render the transfer of modularity explicit, thereby producing infinitely many modular abelian surfaces. This is joint work with Frank Calegari and David Roberts.
10/5/2020 - Koji Shimizu (UC Berkeley) - A p-adic monodromy theorem for de Rham local systems (abstract)

For a family of elliptic curves over a p-adic field, the locus of potentially good reduction defines an analytically open subset of the base. We will explain a generalization of this result to p-adic local systems on a smooth rigid analytic variety by establishing a p-adic monodromy theorem.
9/28/2020 - Rachel Newton (University of Reading) - Evaluating the wild Brauer group (abstract)

The local-global approach to the study of rational points on varieties over number fields begins by embedding the set of rational points on a variety X into the set of its adelic points. The Brauer-Manin pairing cuts out a subset of the adelic points, called the Brauer-Manin set, that contains the rational points. If the set of adelic points is non-empty but the Brauer-Manin set is empty then we say there's a Brauer-Manin obstruction to the existence of rational points on X. Computing the Brauer-Manin pairing involves evaluating elements of the Brauer group of X at local points. If an element of the Brauer group has order coprime to p, then its evaluation at a p-adic point factors via reduction of the point modulo p. For p-torsion elements this is no longer the case: in order to compute the evaluation map one must know the point to a higher p-adic precision. Classifying p-torsion Brauer group elements according to the precision required to evaluate them at p-adic points gives a filtration which we describe using work of Bloch and Kato. Applications of our work include addressing Swinnerton-Dyer's question about which places can play a role in the Brauer-Manin obstruction. This is joint work with Martin Bright.
09/21/2020 - Simon Marshall (University of Wisconsin) - Counting cohomological cusp forms on GL_3 (abstract)

I will give an overview of the limit multiplicity problem for automorphic representations, and discuss some of its applications. I will then present a result in the special case of cohomological forms on GL_3 of fixed level and growing weight. My proof uses p-adic methods of Calegari and Emerton.
09/14/2020 - Preston Wake (Michigan State University) - Tame derivatives and the Eisenstein ideal (abstract)

As was made famous by Mazur, the mod-5 Galois representation associated to the elliptic curve X_0(11) is reducible. Less famously, but also noted by Mazur, the mod-25 Galois representation is reducible. We’ll talk about this kind of extra reducibility phenomenon more generally, for cuspforms of even weight k and prime level. We'll observe that the characters appearing in the reducible representation are related, on one hand, to an algebraic invariant (the ‘tame deriviative’ of an L-function), and, on the other hand, to an algebraic invariant (the 'tame L-invariant'). This type of 'algebraic=analytic' relation is predicted by a version of the Bloch-Kato conjecture for families of motives formulated by Kato.

2019-2020

Seminars for the rest of the year cancelled due to university-wide suspension of non-essential events during coronavirus outbreak.
11/25/2019 - Jiakun Pan (TAMU) - Quantum Unique Equidistribution conjecture for Eisenstein series in the level aspect (abstract)

We study Eisenstein series on growing levels with general central characters, and find an asymptotic formula for their mass distribution, which we call Quantum Unique Ergodicity (QUE). In addition, as an analogy of the t-aspect small scale QUE for Eisenstein series by Young and Humphries separately, we consider above formula under weaker assumptions for the test functions. Consequently, complications occur, to our surprise...
11/18/2019 - Rizwanur Khan (University of Mississippi) - Non-vanishing of Dirichlet L-functions (abstract)

L-functions are fundamental objects in number theory. At the central point s = 1/2, an L-function L(s) is expected to vanish only if there is some deep arithmetic reason for it to do so (such as in the Birch and Swinnerton-Dyer conjecture), or if its functional equation specialized to s = 1/2 implies that it must. Thus when the central value of an L-function is not a "special value", and when it does not vanish for trivial reasons, it is conjectured to be nonzero. In general it is very difficult to prove such non-vanishing conjectures. For example, nobody knows how to prove that L(1/2, \chi) is nonzero for all primitive Dirichlet characters \chi. In such situations, analytic number theorists would like to prove 100% non-vanishing in the sense of density, but achieving any positive percentage is still valuable and can have important applications. In this talk, I will discuss work on establishing such positive proportions of non-vanishing for Dirichlet L-functions.
11/4/2019 - Michał Zydor (University of Michigan) - On the Global Gan-Gross-Prasad conjecture for unitary groups (abstract)

I will discuss the global Gan-Gross-Prasad conjectures and their refinement due to Ichino-Ikeda in the case of unitary groups. I will present the ongoing work with Chaudouard on the relative trace formula (RTF) of Jacquet-Rallis which leads to new results. I will also discuss an extension of these conjectures to non-tempered packets that results from analysis of the fine spectral expansion of the RTF.

2018-2019

06/17/2019 - (Joint seminar with ring theory) Sudesh K. Khanduja (IISER Mohali) - A walk through integrally closed domains and their applications in Number Theory (abstract)

Let $R$ be an integrally closed domain and $\theta$ be an element of an integral domain containing $R$ with $\theta$ integral over $R$ and $F(x)$ be the minimal polynomial of $\theta$ over the quotient field of $R$ . It is an important problem to determine some necessary and sufficient criterion to be satisfied by $F(x)$ so that $R[\theta]$ is an integrally closed domain. This problem was initiated by Dedekind in 1878. In this lecture, we discuss such a criterion when $R$ is a valuation ring. We shall also give some applications of this criterion for algebraic number fields and derive necessary and sufficient conditions involving only the primes dividing $a,b,m,n$ for $\mathbb{Z}[\theta]$ to be integrally closed when $\theta$ is a root of an irreducible trinomial $x^n +ax^m +b$ with coefficients from the ring $\mathbb{Z}$ of integers. Our results led us to prove in 2017-18 the converse of a well known theorem of Algebraic Number theory which says that if $K_1,K_2$ are algebraic number fields with coprime discriminants, then $K_1, K_2$ are linearly disjoint over the field of rational numbers and $A_{K_1} A_{K_2}$ is integrally closed, $A_{K_i}$ being the ring of algebraic integers of $K_i$ . The converse will be discussed in a more general set up of arbitrary valution rings.
04/29/2019 - Jay Jorgenson (CCNY) - Construction of non-holomorphic Eisenstein-type series and their Kronecker limit formulas (abstract)

We will describe a means by which one can define and study a generalization of the non-holomorphic "elliptic" Eisenstein series from PSL(2,R). We prove that our generalization admits a meromorphic continuation and a type of Kronecker limit function. As an example, we consider n-dimensional projective space and show that our approach leads to new expressions for Mahler measures of linear forms in terms of convergent series expansions. This work is joint with James Cogdell and Lejla Smajlovic.
04/17/2019 - (Wednesday 4:15--5:15, Joint seminar with representation theory) Joachim Schwermer (Universität Wien) - On the general linear group over arithmetic orders, related automorphic representations and corresponding cohomology groups (abstract)

Orders in finite-dimensional algebras over number fields give rise to interesting locally symmetric spaces and algebraic varieties. Hilbert modular varieties or arithmetically defined hyperbolic 3-manifolds, compact ones as well as non-compact ones, are familiar examples. In this talk we discuss various cases related to the general linear group GL(2) over orders in division algebras of degree d defined over some number field. The underlying algebraic group is an inner form of the k-split group GL(2d). Geometry, arithmetic, and the theory of automorphic forms are interwoven in a most fruitful way in this work. We indicate a construction of automorphic representations that represent non-vanishing square-integrable cohomology classes for such arithmetically defined groups.
04/15/2019 - Alia Hamieh (University of Northern British Columbia) - Value-Distribution of Cubic Hecke $L$ -functions (abstract)

In this talk, we survey some recent results on the distribution of values of various families of $L$ -functions in the critical strip. We also describe a value-distribution theorem for the logarithms and logarithmic derivatives of a family of $L$ -functions attached to cubic Hecke characters. As a corollary we establish the existence of an asymptotic distribution function for the error term of the Brauer-Siegel asymptotic formula for a certain family of cubic extensions of $\mathbb{Q}(\sqrt{-3})$ . We also deduce a similar result for the Euler-Kronecker constants of this family. This is joint work with Amir Akbary.
02/25/2019 - Harald Helfgott (Gottingen and CNRS) - Summing $\mu(n)$ : a better elementary algorithm (abstract)

Consider either of two related problems: determining the precise number $\pi(x)$ of prime numbers $p\leq x$ , and computing the Mertens function $M(x) = \sum_{n\leq x} \mu(n)$ , where $\mu$ is the M\"obius function. The two best algorithms known are the following: An analytic algorithm (Lagarias-Odlyzko, 1987), with computations based on integrals of $\zeta(s)$ ; its running time is $O(x^{1/2+\epsilon})$ . A more elementary algorithm (Meissel-Lehmer, 1959 and Lagarias-Miller-Odlyzko, 1985; refined by Del\'eglise-Rivat, 1996), with running time about $O(x^{2/3})$ . The analytic algorithm had to wait for almost 30 years to receive its first rigorous, unconditional implementation (Platt), which concerns only the computation of $\pi(x)$ . Moreover, in the range explored to date ( $x\leq 10^{24}$ ), the elementary algorithm is faster in practice. We present a new elementary algorithm with running time about $O(x^{3/5})$ for computing $M(x) = \sum_{n\leq x} \mu(n)$ . The algorithm should be adaptable to computing $\pi(x)$ and other related problems. (joint with L. Thompson)
01/28/2019 - Lola Thompson (Oberlin) - Counting quaternion algebras with applications to spectral geometry (abstract)

We discuss how classical techniques from analytic number theory can be used to count quaternion algebras over number fields subject to various constraints. Because of the correspondence between maximal subfields of quaternion algebras and geodesics on arithmetic hyperbolic manifolds, these counts have interesting applications to the field of spectral geometry. This talk is based on a joint paper with B. Linowitz, D. B. McReynolds, and P. Pollack.
01/14/2019 - Gene Kopp (Bristol) - From Hilbert's 12th problem to complex equiangular lines (abstract)

We describe a connection between two superficially disparate open problems: Hilbert's 12th problem in number theory and Zauner's conjecture in quantum mechanics and design theory. Hilbert asked for a theory giving explicit generators of the abelian Galois extensions of a number field; such an "explicit class field theory" is known only for the rational numbers and imaginary quadratic fields. Zauner conjectured that a configuration of d^2 pairwise equiangular complex lines exists in d-dimensional Hilbert space (and additionally that it may be chosen to satisfy certain symmetry properties); such configurations are known only in a finite set of dimensions d. We prove a conditional result toward Zauner's conjecture, refining insights of Appleby, Flammia, McConnell, and Yard gleaned from the numerical data on complex equiangular lines. We prove that, if there exists a set of real units in a certain abelian extension of a real quadratic field (depending on d) satisfying certain congruence conditions and algebraic properties, a set of d^2 equiangular lines may be constructed, when d is an odd prime congruent to 2 modulo 3. We give an explicit analytic formula that we expect to yield such a set of units. Our construction uses values of derivatives of zeta functions at s = 0 and is closely connected to the Stark conjectures over real quadratic fields. We will work through the example d = 5 in detail to help illustrate our results and conjectures.
11/05/2018 - Keshav Aggarwal (OSU) - Application of trivial delta method towards subconvexity (abstract)

A delta method is an analytic technique to detect when two numbers are equal. Various versions of the delta method have been applied to achieve subconvex bounds for GL(1), GL(2) and GL(3) L-functions (and their Rankin-Selberg convolution) in t, level and weight aspect. It is natural to explore the simplification of these techniques to better understand the crux of the proofs, and possibly extend or improve the known results. We will discuss applications of a 'trivial' delta method in this regard.
10/22/2018 - Elad Zelingher (Yale University) - Exterior square gamma factors for cuspidal representations of $\mathrm{GL}_n$ (abstract)

In 1990 Jacquet and Shalika defined a family of local integrals forming an integral representation of the local exterior square $L$ function of a generic irreducible representation of $\mathrm{GL}_n \left( F \right)$ , where $F$ is a local non-archimedean field of characteristic zero. Later Cogdell and Matringe showed that these integrals satisfy functional equations, which allows one to define the local exterior square factors of a generic irreducible representation of $\mathrm{GL}_n \left( F \right)$ . In this talk, we define analogs of the Jacquet-Shalika integrals for irreducible cuspidal representations of $\mathrm{GL}_n \left( \mathbb{F}_q \right)$ , and discuss the functional equations they satisfy. We define the exterior square gamma factor for such representation and express it using the Bessel function associated with the representation. We relate our analogs of the Jacquet-Shalika integrals to the local integrals using level zero representations. If time permits, we will also discuss our work on the local exterior square factors of simple supercuspidal representations. This is joint work with Rongqing Ye.
10/15/2018 - Nicolae Anghel (University of North Texas) - Heron Triangles with Constant Area and Perimeter (abstract)

The talk undertakes a very detailed, very visual, and quite elementary study of the Heron triangles of fixed area and perimeter. It circumvents the traditional approach to Heron triangles based on elliptic curves. Its key focus is on the geometry, calculus, and algebra of the associated area curve. The main result presents a simple sufficient condition for the existence of infinitely many Heron triangles with constant area and perimeter. An application to Diophantine equations is also given.
10/08/2018 - Rongqing Ye (OSU) - Rankin--Selberg gamma factors of level zero representations of $GL_n$ (abstract)

For a $p$ -adic local field $F$ of characteristic 0, with residue field $\mathfrak{f}$ , I will show in the talk that the Rankin--Selberg gamma factor of a pair of level zero representations of linear general groups over $F$ is equal to a gamma factor of a pair of corresponding cuspidal representations of linear general groups over $\mathfrak{f}$ . This results can be used to prove a variant of Jacquet's conjecture on local converse theorem.
09/17/2018 - Bill Mance (Adam Mickiewicz University) - Normal numbers with respect to the Cantor series expansions and possible applications to studying algebraic varieties (abstract)

We will discuss basic properties of normal numbers for the Cantor series expansions and a recent result of D. Airey and B. Mance. Using ideas introduced in this paper as well as techniques from descriptive set theory, it may be possible to show that information about algebraic varieties is encoded in the structure of sets of normal numbers. We will outline this idea and the barriers one may encounter in finishing it.

2017-2018

03/19/2018 - Andrew Obus (UVA) - Reduction of dynatomic curves: The good, the bad, and the irreducible (abstract)

The dynatomic modular curves parameterize one-parameter families of dynamical systems on P^1 along with periodic points (or orbits). These are analogous to the standard modular curves parameterizing elliptic curves with torsion points (or subgroups). For the family x^2 + c of quadratic dynamical systems, the corresponding modular curves are smooth in characteristic zero. We give several results about when these curves have good/bad reduction to characteristic p, as well as when the reduction is irreducible. We will also explain some motivation from the uniform boundedness conjecture in arithmetic dynamics, coming from results of Poonen, Flynn, Schafer, Stoll, and Morton.
03/05/2018 - Qing Zhang (Sun Yat-Sen University) - Local converse theorem for unitary groups (abstract)

Local converse theorem says that an irreducible generic representation of a classical group over a p-adic field should be determined by its various local gamma factors twisting with GL_n. In this talk, I will describe a sketch of a proof of local converse theorems for quasi-split unitary groups. A main ingredient of the proof is certain properties of partial Bessel functions developed by Cogdell-Shahidi-Tsai.
02/26/2018 - Joseph Vandehey (OSU) - Ergodicity of geometrically complete Iwasawa continued fractions (abstract)

While multiple proofs of ergodicity exist for the standard real continued fractions, proofs of ergodicity for complex continued fractions rely on a serendipitous decomposition such that even a slight permutation would destroy the proof. Continuing from a talk given several years ago, we establish a more general framework of the connection between continued fractions and hyperbolic space, which allows us to prove ergodicity in many more interesting cases, including on the Heisenberg group, where it has been an open question.
01/29/2018 - Edmund Karasiewicz (UC Santa Cruz) - The Fourier Coefficients of a Minimal Parabolic Eisenstein Series on the Double Cover of GL(3) over Q (abstract)

Eisenstein series are special types of automorphic forms that have proved useful in the study of L-functions. For example, Shimura considered a Rankin-Selberg construction using an Eisenstein series on the double cover of SL(2) to study the symmetric square L-function of a modular form. We will discuss the Fourier coefficients of the titular Eisenstein series and describe some potential applications to the Archimedean theory of GL(3) symmetric square L-functions and the study of moments of quadratic Dirichlet L-functions.
11/27/2017 - Yeongseong Jo (OSU) - The local exterior square L-function for GL(n) (abstract)

In mid 1990's Cogdell and Piatetski-Shapiro embarked a project to compute the local exterior square L-functions for irreducible admissible generic representations of GL(n). In this talk I describe how one can express those L-functions in terms of symmetric or exterior square L-functions for the inducing datum. The main two ingredients for this computation are exceptional poles and the method of derivatives of Bernstein and Zelevinsky. Time permitting, I explain that the exterior square arithmetic (Artin) and analytic L-functions for GL(n) through integral representations coincide.
11/20/2017 - Bob Hough (Stony Brook) - The shape of cubic and quartic fields (abstract)

I introduce a new zeta function on the space of binary cubic forms, and the space of pairs of ternary quadratic forms, which is twisted by a cusp form. I show that these zeta functions are entire. Combined with a method of Taniguchi-Thorne, I obtain equidistribution of the shape of cubic fields, and of the joint shape of a quartic field and it's cubic resolvent ring, when fields are ordered by discriminant. Here the shape of a number field is understood to be the shape of the ring of integers viewed as a lattice in the canonical embedding, and projected in the direction orthogonal to 1.
11/13/2017 (3pm in MW154) - Claire Burrin (Rutgers) - Dedekind sums for cofinite Fuchsian groups (abstract)

Dedekind sums, studied since the late 19th century, are closely related to an amazing variety of objects ranging through combinatorics, geometry, number theory, and physics. Among those, we simply mention the discriminant form from the classical theory of modular forms, and winding and linking numbers for modular geodesics. For each cusp of a cofinite Fuchsian group, there is a natural construction from the theory of automorphic forms that can be used to define generalized Dedekind sums. We will show that these generalized Dedekind sums obey certain reciprocity laws, and discuss their relation to winding numbers for closed geodesics on hyperbolic surfaces.
10/23/2017 - Keshav Aggarwal (OSU) - Hybrid subconvex bound for certain Hecke character L-functions (abstract)

Let $\psi$ be a Hecke chacter. Kaufman (1979) and Sohne (1996) proved a $t$ -aspect subconvex bound for $L(s,\psi)$ , while recently Michel-Venkatesh (2010) proved a uniform subconvex bound in all aspects. Wu (2016) followed Michel-Venkatesh to prove a conditional Burgess bound in conductor aspect. We follow recent methods developed by Munshi to prove a Weyl bound in $t$ -aspect for Hecke character L-functions of imaginary quadratic fields, and a hybrid bound in conductor aspect as strong as the Weyl bound.
10/16/2017 - Yongxiao Lin (OSU) - Bounds for twists of $\rm GL(3)$ $L$ -functions (abstract)

In this talk, we will discuss nontrivial estimates for certain degree 3 $L$ -functions on the critical line. The $L$ -functions we are particularly interested in are $L(s,\pi)$ and $L(s,\pi\otimes \chi)$ , where $\pi$ is a fixed Hecke-Maass cusp form for $\rm{SL}(3,\mathbb{Z})$ and $\chi$ is a primitive Dirichlet character of conductor $q$ (which we assume to be a prime). We will describe our work in establishing $t$ -aspect subconvex bound for $L\left(1/2+it,\pi \right)$ , and the conductor and $t$ aspect subconvex bound for $L(1/2+it, \pi\otimes \chi)$ , under the assumption $q^{\varepsilon} < |t| < q^{34/21}$ . Time permitting, we shall give a sketch of the proof for a subconvex bound $L\left(1/2+it,\pi \right)\ll (|t|+2)^{3/4-\delta}$ , where $\delta=1/60-\varepsilon$ .
10/09/2017 - Shenhui Liu (University of Toronto) - Central L-values of Maass wave forms (abstract)

In this talk, I will focus on central L-values of Maass wave forms and present a positive-proportional nonvanishing result of such values in the aspect of large spectral parameter in short intervals, which is qualitatively optimal in view of Weyl's law. The result is obtained by studying mollified moments with the Kuznetsov trace formula (for the first moment) and a formula of Motohashi (for the second moment). As an application of this result and a formula of Katok--Sarnak, I will give a nonvanishing result of the first Fourier coefficients of Maass forms of weight 1/2 and level 4 in the Kohnen plus space.
10/02/2017 - Roman Holowinsky (OSU) - Subconvexity for twists of automorphic L-functions without the delta method (abstract)

In this talk, we demonstrate how to simplify some of the recent proofs of R. Munshi which make use of a Petersson trace formula delta method. We show that the delta method may actually be removed altogether in certain applications and instead replaced with appropriate dual summation formulae. The case we shall discuss in detail is subconvexity for Dirichlet character twists of GL(3) automorphic L-functions. This is joint work with Paul Nelson.

2016-2017

06/01/2017 (Thursday 12.40pm in MW154) - Paul Nelson (ETH Zurich) - Subconvex equidistribution of cusp forms (abstract)

Arithmetic quantum chaos concerns the limiting behavior of a sequence of automorphic forms on spaces such as the modular surface. It is now known in many cases (by work of Lindenstrauss, Holowinsky, Soundararajan and others) that the mass distributions of such forms equidistribute as the parameters tend off to infinity. Unfortunately, the known rates of equidistribution are typically weak (ineffective or logarithmic in the parameters). I will discuss the problem of obtaining strong rates (power savings) and the related subconvexity problem, emphasizing recent progress on the level aspect.
04/24/2017 - Baiying Liu (Purdue) - On the local converse theorem for GL $_n$ (abstract)

In this talk, I will introduce a complete proof of a standard conjecture on the local converse theorem for generic representations of $GL_n(F)$ , where $F$ is a non-archimedean local field. This is a joint work with Prof. Herve Jacquet (see arXiv:1601.03656). I will also briefly talk about extensions of local converse theorem to the setting of l-adic families and modular l representations, which is a joint work with Gilbert Moss.
04/17/2017 - Bao V. Le Hung (U Chicago) - Congruences between automorphic forms (abstract)

The theory of congruences between automorphic forms traces back to Ramanujan, who observed various congruence properties between coefficients of generating functions related to the partition function. Since then, the subject has evolved to become a central piece of contemporary number theory; lying at the heart of spectacular achievements such as the proof of Fermat's Last Theorem and the Sato-Tate conjecture. In my talk I will explain how the modern theory gives satisfactory explanations of some concrete congruence phenomena for modular forms (the GL_2 case), as well as recent progress concerning automorphic forms for higher rank groups. This is joint work with D. Le, B. Levin and S. Morra.
04/03/2017 - Liyang Zhang (Yale University) - Quantum Unique Ergodicity of Degenerate Eisenstein Series on GL(n) (abstract)

In the area of quantum chaos, it is of great interest to investigate the distribution of the $L^2$ -mass of the eigenfunctions of the Laplacian as the eigenvalues tend to infinity. Luo and Sarnak first formulated and proved quantum unique ergodicity of Eisenstein series on SL $(2, \mathbb{Z})$ \ $\mathcal{H}$ . In this talk, we extend the result of Luo and Sarnak and prove quantum unique ergodicity for a subspace of the continuous spectrum spanned by the degenerate Eisenstein Series on GL(n).
03/06/2017 - Kęstutis Česnavičius (UC Berkeley) - The Manin constant in the semistable case (abstract)

For an optimal modular parametrization $J_0(n) \twoheadrightarrow E$ of an elliptic curve $E$ over $\mathbb{Q}$ of conductor $n$ , Manin conjectured the agreement of two natural $\mathbb{Z}$ -lattices in the $\mathbb{Q}$ -vector space $H^0(E, \Omega^1)$ . Multiple authors generalized his conjecture to higher dimensional newform quotients. We will discuss the semistable cases of the Manin conjecture and of its generalizations using a technique that establishes general relations between the integral $p$ -adic etale and de Rham cohomologies of abelian varieties over $p$ -adic fields.
02/27/2017 - Jay Jorgenson (CCNY) - Modular Dedekind symbols associated to Fuchsian groups and higher-order Eisenstein series (abstract)

Higher-order Eisenstein series are defined as convergent series which involve periods of a weight two holomorphic form. Associated to each such series, we study its corresponding Dedekind symbol. In certain cases when the underlying group is arithmetic, we prove that the Dedekind symbols are rational numbers. In all cases, we prove various relations including a reciprocity law. The work is joint with Cormac O'Sullivan and Lejla Smajlovic.
02/20/2017 - Chen Wan (UMN) - Multiplicity one theorem for the Ginzburg-Rallis model (abstract)

Following the method developed by Waldspurger and Beuzart-Plessis in their proof of the local Gan-Gross-Prasad conjecture, we were able to prove the multiplicity one theorem on Vogan L-packet for the Ginzburg-Rallis model. In some cases, we also proved the epsilon dichotomy conjecture which gives a relation between the multiplicity and the value of the exterior cube epsilon factor.
02/13/2017 - Ralf Schmidt (Univ. of Oklahoma) - The paramodular conjecture and the representation theory of GSp(4) (abstract)

In 2014 Brumer and Kramer formulated the "Paramodular Conjecture", which is a degree 2 version of Shimura-Taniyama-Weil. One version of the conjecture states that the L-function of an abelian surface defined over $\mathbb{Q}$ with conductor $N$ is also the L-function of a Siegel modular form of degree 2 with respect to the paramodular group of level $N$ . In this talk we will explain, using the local representation theory of the group GSp(4), why it is natural to expect paramodular forms to appear in this conjecture.
01/30/2017 - Brad Rodgers (Univ. Michigan) - Sums in short intervals and decompositions of arithmetic functions (abstract)

In this talk we will discuss some old and new conjectures about the behavior of sums of arithmetic functions in short intervals, along with analogues of these conjectures in a function field setting that have been proved in recent years. We will pay particular attention to some surprising phenomena that comes into play, and a decomposition of arithmetic functions in a function field setting that helps elucidate what's happening.
01/23/2017 - Eric Katz (OSU) - Tropical geometry and torsion point bounds (abstract)

We discuss how tropical geometry techniques can be used to bound the number of torsion points on algebraic curves. We describe our recent progress towards a uniform bound in terms of the first prime of compact type reduction or the first prime of really awful reduction. This is joint work with Taylor Dupuy, Joseph Rabinoff, and David Zureick-Brown.
10/10/2016 - Shenhui Liu (OSU) - Central L-derivative values of automorphic forms (abstract)

In this talk I will present results on average behaviors and nonvanishing of the central $L$ -derivative values of $L(s,f)$ and $L(s,f_{K_D})$ for $f$ in an orthogonal Hecke eigenbasis $H_{2k}$ of weight $2k$ cusp forms of level 1 for large odd $k$ . Here $f_{K_D}$ is the base change of $f$ to an imaginary quadratic field $K_D=\mathbb{Q}(\sqrt{D})$ with fundamental discriminant $D$ . I will also give applications of such results to Heegner cycles of high weights of the modular curve $X_0(1)$ .
09/26/2016 - Joseph Vandehey (OSU) - When multiplying by $2$ is a hard thing to do (abstract)

Continued fractions react in surprisingly complex fashion to simple operations like multiplying by $2$ or adding $1/3$ . While we have algorithms that allow us to handle such things computationally, it's not clear how these operations affect continued fractions from a theoretical framework. We will show that the property of normality is preserved under these operations, extending a famous result of Wall. There will be substantial overlap with ergodic theory.

2015-2016

04/25/2016 - Riad Masri (Texas A&M) - Singular moduli and the distribution of partition ranks (abstract)

In this talk I will discuss some problems concerning the distribution of partition ranks among residue classes and explain how they can be resolved in certain cases using methods from the spectral theory of automorphic forms.
04/18/2016 - Jonathan Bober (Bristol) - Computing modular forms (abstract)

I'll describe how to use the trace formula to compute spaces of classical modular forms (in weights >= 2) and how to get information about weight 1 spaces, and some computations I've done recently and have planned. This includes, for example, dimensions of spaces of weight 1 forms for all levels < 2000 and all characters, and practical numerical computation of Fourier expansions of newforms with general character.
04/14/2016 (Thursday 4.30pm in MW154) - Hongze Li (Shanghai Jiao Tong University) - Small gaps between the Piatetski-Shapiro primes (abstract)

There are many progress in the small gaps between primes in recent years, especially, Yitang Zhang proved that there are infinitely many couples of primes with the difference no more than 70000000, J.Maynard proved that the 70000000 can be repleced by 600. The best result is 246 due to polymath. In this talk, I shall discuss the small gaps between the Piatetski-Shapiro primes which is a special kind of primes and forms a thin set of primes, and I shall show the following result for the small gaps between the Piatetski-Shapiro primes. Suppose that $1 < c < 9/8$ . For any $m\geq 1$ , there exist infinitely many $n$ such that $\{[n^c],\ [(n+1)^c],\ \ldots,\ [(n+k_0)^c]\}$ contains at least $m+1$ primes, if $k_0$ is sufficiently large (only depending on $m$ and $c$ ). This is a joint work with Dr. Hao Pan.
04/04/2016 - Pankaj Vishe (Durham) - Hasse principle for higher degree hypersurfaces (abstract)

A projective variety X defined over the rational numbers is said to satisfy the Hasse principle if the presence of an adelic point on X guarantees the presence of a rational point. We give a programme for establishing Hasse principle for hypersurfaces of degree at least 4. The key ingredient is Kloosterman type extra averaging in conjunction with the van der Corput differencing applied to estimate the ''minor arc contribution'' in the Hardy-Littlewood circle method. We show the utility of this approach by improving upon current bounds in the quartic case. This is a joint work with Oscar Marmon.
03/28/2016 - Jack Buttcane (SUNY) - The Kuznetsov formula on GL(3) (abstract)

The GL(2) Kuznetsov formula gives a connection between Kloosterman sums and Fourier coefficients of Maass forms. I will discuss its generalization to GL(3) and applications to the theory of exponential sums and GL(3) L-functions.
03/23/2016 (Wednesday 4.30pm in MW154)- Leo Goldmakher (Williams) - Characters and their nonresidues (abstract)

Understanding the least quadratic nonresidue (mod p) is a classical problem, with a history stretching back to Gauss. The approach which has led to the strongest results uses character sums, objects which are ubiquitous in analytic number theory. I will discuss character sums, their connection to the least nonresidue, and work of myself and Jonathan Bober (University of Bristol) on a new approach to the problem.
03/07/2016 - Aaron Pollack (Stanford) - The Spin $L$ -function on GSp $(6)$ (abstract)

I will discuss a Rankin-Selberg integral that may be thought of as a close cousin of the triple product integral of Garrett and Piatetski-Shapiro--Rallis. The integral unfolds to the degree $8$ Spin $L$ -function of cuspidal automorphic representations of PGSp $(6)$ . When the automorphic representation corresponds to a level one Siegel modular form, we deduce the finiteness of poles and functional equation of the completed L-function. The arithmetic invariant theory of quaternion algebras and their orders plays an important role.
02/22/2016 - Padmavathi Srinivasan (MIT) - Conductors and minimal discriminants of hyperelliptic curves with rational Weierstrass points (abstract)

Conductors and minimal discriminants are two measures of degeneracy of the singular fiber in a family of hyperelliptic curves. In the case of elliptic curves, the Ogg-Saito formula shows that (the negative of) the Artin conductor equals the minimal discriminant. In the case of genus two curves, equality no longer holds in general, but the two invariants are related by an inequality. We investigate the relation between these two invariants for hyperelliptic curves of arbitrary genus.
02/15/2016 - Sheng-Chi Liu (WSU) - The distribution of integral points on homogeneous varieties (abstract)

In this talk we will give a broad overview of the Linnik problems concerning the equidistribution of integral points on homogeneous varieties. One particular example concerns the Heegner points, which are roots in the complex upper-half plane of certain quadratic forms. We will discuss certain "sparse" equidistribution problems concerning these points and give an application of an analog of Linnik's famous theorem on the first prime in an arithmetic progression.
The resolution of these problems involves a wide-range of techniques concerning modular forms and their associated L-functions.
This is joint work with Riad Masri and Matt Young.
01/25/2016 - Scott Ahlgren (UIUC) - Kloosterman sums and Maass cusp forms of half integral weight for the modular group (abstract)

Kloosterman sums appear in many areas of number theory. We estimate sums of Kloosterman sums of half-integral weight on the modular group. Our estimates are uniform in all parameters in analogy with Sarnak and Tsimerman's improvement of Kuznetsov's bound for the ordinary Kloosterman sums. Among other things we require mean value estimates for coefficients of Maass cusp forms of half integral weight and uniform estimates for K-Bessel integral transforms. As an application, we obtain an improved estimate for the classical problem of estimating the size of the error term in Rademacher's famous formula for the partition function. This is joint work with Nick Andersen.
11/23/2015 - Ali Altuğ (Columbia University) - Arthur-Selberg trace formula and related problems (abstract)

I will report on recent research on the Arthur-Selberg trace formula for $GL(2)$ . The aim is to understand the internal structure of the trace formula in a way that would allow to use it directly in non-comparative analytic applications. The original motivation for this comes from Langlands' ``Beyond Endoscopy'', which aims to understand the fine structure of the automorphic spectrum of a group. If time permits I will also discuss several problems and other implications of the same circle of ideas.
11/02/2015 - Jay Jorgenson (CCNY) - Recent results in the study of groups of moonshine-type (abstract)

We will present results obtained in studying the spectral theory and holomorphic function theorem associated to the groups denote by $\Gamma_{0}(N)^{+}$ , which are obtained by "adding" the Fricke involutions to the principal congruence groups. If $N$ is square-free and the associated quotient space has genus zero, then the groups arise in the theory of "monstrous moonshine". In addition to theoretical considerations, we employed extensive computer programs in order to, for example, determine canonical generators of the function fields for all genus one, two and three groups. The work is joint investigations with Lejla Smajlovic and Holger Then.
09/21/2015 - Ahmad El-Guindy (Texas A&M University at Qatar) - On Power Eigensystems of Drinfeld-Goss Hecke eigenforms (abstract)

We shall discuss some remarkable properties of the Hecke action on (doubly cuspidal) modular forms in the Drinfeld function field setting; highlighting some of the striking differences with the classical case. We present new results on some common properties of certain coefficients of such forms that depend only on the eigenvalue (but not the weight), as well as a precise conjecture regarding the distribution of weights corresponding to such eigen systems.

2014-2015

04/13/2015 - Baiying Liu (Utah) - On cuspidality of global Arthur packets of quasi-split classical groups (abstract)

Based on the theory of endoscopy, Arthur classified the automorphic discrete spectrum of quasi-split classical groups up to global Arthur packets parametrized by Arthur parameters. Towards studying representations in each Arthur packet, a natural question one may ask is that whether a given Arthur packet has cuspidal representations or not. Historically, this question has close relation to the theory of singular automorphic forms, which has been investigated by Roger Howe and many others. In this talk, I will introduce some recent progress on this aspect, which is based on relations between the structure of Fourier coefficients of automorphic forms in an Arthur packet and the structure of the corresponding Arthur parameter. This work is joint with Dihua Jiang.
03/30/2015 - Kate Petersen (FSU) - Equidistribution of Norm 1 elements in Number Fields (abstract)

In a natural way, one can view the norm 1 elements in a number field K as lying in a torus, upon quotienting by the group of units. I'll discuss the problem of determining if the set of norm 1 elements is equidistributed with respect to an ordering by visible points, proving this in the case when K is Galois with cyclic Galois group. This is joint work with Chris Sinclair.
03/24/2015 (Tuesday 4.30pm in CH240) - Guangshi Lü (Shandong University) - Average behaviour of certain arithmetic functions (abstract)

In this talk, we shall introduce some results on long interval (or short interval) asymptotic fomulae for ceratin arithmetic functions. In particular, we shall talk about the average behaivor of Fourier coefficients of automorphic forms, the distribution of integers having a given number of prime factors in short intervals, etc.
03/23/2015 - Nigel Pitt (Maine) - A cuspidal analogue of Titchmarsh's divisor problem (abstract)

Given the Fourier coefficients $a(n)$ of a holomorphic cusp form for the modular group, we will show that $\sum_{\substack{p\le X\\p \textrm{ prime}}} a(p-1) \ll X^{391/392+\epsilon}$ for any $\epsilon >0$ for large $X$ . Similarly $\sum_{n\le X} \mu(n)a(n-1) \ll X^{391/392+\epsilon}.$ We will sketch the proofs, which require establishing non-trivial bounds for sums of Kloosterman sums and shifted convolutions of the coefficients which are better in the ranges required than known estimates. These are then used to bound bilinear forms in $a(m n - 1)$ , which in conjunction with previous work of the speaker proves the main results.
03/09/2015 - Paul Pollack (UGA) - Two analytic problems on CM elliptic curves (abstract)

I will discuss two problems about CM elliptic curves. The first deals with statistics of their reductions modulo a prime, as the prime varies. Suppose $E/Q$ is a fixed CM elliptic curve. For each prime $p$ of good reduction, write $E(F_p) = Z/d_p Z \oplus Z/e_p Z$ , where $d_p$ divides $e_p$ . So, for example, $d_p=1$ iff $E(F_p)$ is cyclic. Kowalski asked for estimates on the average size of $d_p$ . We discuss recent work with T. Freiberg showing that $\sum_{p \le x} d_p \asymp x$ . The second project concerns torsion structures on CM elliptic curves over number fields. For each positive integer $d$ , let $T_{CM}(d)$ denote the largest possible size of a torsion subgroup of an elliptic curve defined over a degree $d$ number field. I will discuss recent work with Clark where we determine the upper asymptotic order of $T_{CM}(d)$ , as a function of $d$ . Time permitting, I will mention recent results with Bourdon and Clark that shed light on the typical size of $T_{CM}(d)$ .
02/09/2015 - Wissam Raji (AUB) - Unimodularity of zeros of period polynomials of Hecke Eigenforms (abstract)

We prove that all zeros of the full period function of any Hecke Eigenforms lie on the unit circle.
11/24/2014 - Dani Szpruch (Indiana) - The Langlands Shahidi method and the metaplectic exceptional representations (abstract)

The Langlands-Shahidi method has proven to be one of the most powerful tools to study automorphic L-functions. In this talk we shall survey the definition of Shahidi local coefficients and their application to local reducibility questions. We shall then present some new results and techniques regarding their metaplectic counterpart. Finally we shall discuss the connection between these metaplectic local coefficients and Kazhdan-Patterson's metaplectic exceptional representations. This is a joint work with Solomon Friedberg and David Goldberg.
11/10/2014 - Ian Whitehead (UMN) - Toward a Casselman-Shalika Formula for Metaplectic Kac-Moody Groups (abstract)

I will construct a Dirichlet series in several variables satisfying an infinite group of functional equations, the Weyl group of an affine Kac-Moody Lie algebra. This series can be understood as a generalization of the Casselman-Shalika formula for Whittaker coefficients of Eisenstein series--in this case, Eisenstein series on a metaplectic cover of a Kac-Moody group. It has applications to moment problems in analytic number theory.
11/03/2014 - Eyal Kaplan (OSU) - The theta-period of a cuspidal automorphic representation of GL(n) (abstract)

Let E be an Eisenstein series corresponding to an automorphic cuspidal representation of GL(n), induced to a representation of SO(2n+1) through the Siegel parabolic subgroup. The presence of a pole of E at 1/2, that is, the nonvanishing of the residue E_{1/2} of E at 1/2, is determined by the presence of a pole of the partial symmetric square L-function at 1. We define a co-period integral of E, involving the integrationof E_{1/2} against a pair of automorphic forms in the space of the small representation of Bump, Friedberg and Ginzburg. We compute the co-period and relate it to a ``theta period" integral of GL(n) - an integral of a cusp form against two theta functions corresponding to the exceptional representation of Kazhdan and Patterson.
10/27/2014 - Matthew Young (Texas A&M) - Weyl-type hybrid subconvexity bounds for twisted L-functions and Heegner points on shrinking sets (abstract)

One of the major themes of the analytic theory of automorphic forms is the connection between equidistribution and subconvexity. An early example of this is the famous result of Duke showing the equidistribution of Heegner points on the modular surface, a problem that boils down to the subconvexity problem for the quadratic twists of Hecke-Maass L-functions. It is interesting to understand if the Heegner points also equidistribute on finer scales, a question that leads one to seek strong bounds on a large collection of central values. Work of Conrey and Iwaniec from 2000 gives the best-known subconvexity bound for twisted L-functions, but lacks the uniformity required for the more advanced equidistribution problems. I will discuss recent work that resolves these problems.
10/20/2014 - Min Lee (Bristol) - Shifted multiple Dirichlet series and spectral moments of Rankin-Selberg L-functions (abstract)

In this talk, we develop certain aspects of the theory of shifted multiple Dirichlet series and study their meromorphic continuations. Their analytic properties are used to obtain explicit spectral second moment formulas for Rankin-Selberg L-functions of automorphic forms with applications. This is a joint work with Jeff Hoffstein.
10/13/2014 - Meng Zhang (OSU) - Some improvements on Waring-Goldbach problem for fourth powers with almost equal variables (abstract)

The Waring-Goldbach problem concerns representations of positive integers by powers of primes. The representation with almost equal variables is the more restricted problem which has been studied for many years. In this talk, I will try to use the Hardy-Littlewood circle method to show this problem for fourth powers. This is joint work with Yanjun Yao.
10/06/2014 - Sandro Bettin (CRM) - The twisted second moment of the Dirichlet L-functions (abstract)

We will discuss some properties of the twisted second moment of the Dirichlet L-functions. In particular, we will investigate the reciprocity formula studied by Conrey and Young, highlighting a surprising continuity property. We will also highlight the relation of the twisted moment with the Estermann function, and give the asymptotic for "the moments of the twisted moments".

2013-2014

04/21/2014 - Michael Rubinstein (Waterloo) - Moments of zeta functions associated to hyperelliptic curves (abstract)

I will discuss conjectures, theorems, and experiments concerning the moments, at the central point, of zeta functions associated to hyperelliptic curves over finite fields. This is joint work with Kevin Wu.
04/07/2014 - Paul Bourgade (IAS and Harvard) - Strong Szegö's theorem for $L$ functions (abstract)

Fluctuations of random matrix theory type have been known to occur in analytic number theory since Montgomery's calculation of the pair correlation of the zeta zeros, in the microscopic regime. At the mesoscopic scale, the analogy still holds, through a limiting Gaussian field, restriction of the free field to a line. In particular we will present an unconditional proof for an analogue of the strong Szegö theorem, for $L$ -functions.
03/31/2014 - Adam Harper (Cambridge) - Sharp bounds for moments of the Riemann zeta function (abstract)

The Riemann zeta function $\zeta(s)$ has been studied for more than 150 years, but our knowledge about it remains very incomplete. On or near the critical line $\Re(s)=1/2$ , our knowledge is lacking even if we assume the truth of the Riemann Hypothesis. For example, the behaviour of the power moments $\int_0^T |\zeta(1/2+it)|^{2k} dt$ , which is subject to precise conjectures coming from random matrix theory, has resisted most rigorous study until recently.

In this talk I will try to explain work of Soundararajan, which gave nearly sharp upper bounds for the moments of zeta (assuming the Riemann Hypothesis), and also my recent improvement giving sharp upper bounds (assuming the Riemann Hypothesis).
03/24/2014 - Steven J Miller (Williams College) - Closed-form moments in elliptic curve families and low-lying zeros (abstract)

We explore the behavior of zeros near the central point for families of elliptic curves with rank over Q(T) and small conductors. Zeros of L-functions are conjectured to be simple except possibly at the central point for deep arithmetic reasons; these families provide a fascinating laboratory to explore the effect of multiple zeros on nearby zeros. Though theory suggests the family zeros (those we believe exist due to the Birch and Swinnerton-Dyer Conjecture) should not interact with the remaining zeros, numerical calculations show this is not the case; however, the discrepency is likely due to small conductors. We'll mix theory (including lower order terms in the convergence to the Sato-Tate measure) and experiment and see some surprisingly results, which lead us to conjecture that a new random matrix ensemble correctly models the small conductor behavior. Note: parts of all the computations were done with OSU students when the speaker was a postdoc here.
03/17/2014 - Tim Browning (Bristol) - The arithmetic of large-dimensional varieties is easy (abstract)

We use the Hardy-Littlewood circle method to justify the title of the talk. This is joint work with Roger Heath-Brown.
02/24/2014 - Djordje Milićević (Bryn Mawr) - Sub-Weyl subconvexity and short p-adic exponential sums (abstract)

One of the principal questions about L-functions is the size of their critical values. In this talk, we will present our recent subconvexity bound for the central value of a Dirichlet L-function of a character to a prime power modulus, which breaks a long-standing barrier known as the Weyl exponent. We obtain these results by developing a new general method to estimate short exponential sums involving p-adically analytic fluctuations, which can be naturally seen as a p-adic analogue of the method of exponent pairs. We will present the main results of this method and the key points in its development, and discuss the structural relationship between the p-adic analysis and the so-called depth aspect.
02/17/2014 - Micah Milinovich (Ole Miss) - Estimates for the zeros of the Riemann zeta-function via Fourier analysis (abstract)

In this talk I will show how to use certain Beurling-Selberg type majorants and minorants of exponential type in conjunction with the Guinand-Weil explicit formula to study the vertical distribution of the zeros of the Riemann zeta-function. We can use these techniques to prove the sharpest known bounds for the number zeros in a long interval on the critical line (assuming the Riemann hypothesis) and also to study local statistics of zeros (i.e. pair correlation). Our results on pair correlation extend earlier work of P. X. Gallagher and give some evidence for the famous conjecture of H. L. Montgomery. This is based on joint works with Emanuel Carneiro, Vorrapan Chandee, and Friedrich Littmann.
02/10/2014 - Maksym Radziwill (IAS) - Some recent interactions between sieves and L-functions (abstract)

I will discuss some recent work, where methods from sieve theory and L-functions interact.

In the first direction, I will explain joint work with Kannan Soundararajan, where we find an "L-function analogue" of the Brun-Hooley sieve. Our method has several applications but I'll focus specifically on the problem of estimating small moments of L-functions. This particular problem has corollaries for the distribution of the Tate-Shafarevich group of primes twists of an elliptic curve or for the distribution of coefficients of half-integral weight modular forms.

Time permitting, I will also address the opposite direction, namely adapting technique from L-functions to problems of a more sieve theoretic character. In joint work with Kaisa Matomaki we use the idea of a mollifier, in a rather unusual context, to obtain sharp results on the number of sign changes of Fourier coefficients of Hecke eigencuspforms.
02/03/2014 - Joseph Vandehey (UGA) - The unexpected strength of continued fractions on the Heisenberg group (abstract)

Over the years, many different higher-dimensional continued fraction algorithms have been created to generalize the many useful properties of standard one-dimensional continued fractions. Last year, in joint work with A. Lukyanenko, the speaker discovered a continued fraction algorithm on the Heisenberg group that is surprisingly powerful: it appears to be the first time certain classical formulas have found higher-dimensional analogs. We will discuss these expansions, diophantine approximation, periodic continued fractions, and a number of open questions. Will include some discussion of hyperbolic geometry and ergodic theory.
01/08/2014 - Farrell Brumley (Paris 13) - Counting cusp forms by analytic conductor (abstract)

Given a reductive group over a global field, a natural question is to count the number of cuspidal automorphic representations of bounded analytic conductor. The set-up is highly reminiscent of counting rational points of bounded height on certain algebraic varieties. We'll discuss a proper formulation of the automorphic problem and present a few recent results, obtained in collaboration with D. Milicevic, for general linear groups.
11/25/2013 - Rudy Perkins (OSU) - Interpolating evaluation characters and the Anderson generating function for the Carlitz module (abstract)

We introduce a fundamental tool in the construction of abelian, geometric extensions of the field of rational functions in one indeterminate over a finite field - the Carlitz module. We discuss its action on the Tate algebra and how this gives rise to an incredible function - the Anderson generating function for the Carlitz module (AGF). Finally, we shall prove a formula connecting the AGF to an interpolation series for Pellarin's evaluation character with applications to Taelman's ``unit'' module and explicit formulas for both the rational special values of Pellarin's series and a subset of his deformations of Drinfeld modular forms.
11/18/2013 - Wing Chung (Jonathan) Lam (OSU) - Second moment of the central values of the symmetric square L-functions (abstract)

In this talk we explain how to establish a new bound for the second moment for the central values of the symmetric square L-functions attached to holomorphic forms of large weights. The bound obtained is sharp and improves substantially upon previously known results.
11/04/2013 - Zhilin Ye (OSU) - The Second Moment of Rankin-Selberg L-function and Hybrid Subconvexity Bound (abstract)

Assume that $f$ and $g$ are both holomorphic modular forms with fixed weights over $GL(2)$ . We give a sharp estimation of $\sum |L(1/2, f\times g)|^2$ which is an improvement of the result in Kowalski-Michel-Vanderkam. As a consequence, we prove a level aspect hybrid subconvexity bound for tensor product $f \times g$ .
10/14/2013 - Ghaith Hiary (OSU) - Detecting squarefree numbers (abstract)

Let $d = l^2 \Delta$ , where $\Delta$ is square-free, and $\Delta$ and $l$ are unknown to us. A method to obtain a lower bound on $\Delta$ without attempting to factor $d$ is presented. If $d$ happens to be square-free, then the method might yield a sufficiently good lower bound on $\Delta$ so that the square-freeness of $d$ can be certified fast -- in particular, faster than could have been done had we immediately applied one of the other known methods that also can produce partial information about $\Delta$ (such as the Pollard-Strassen algorithm). The running time of the method is heuristically sub-exponential in the lower bound over a relatively wide initial range, and perhaps further. The method is based on the explicit formula for the Dirichlet $L$ -function associated with a suitably chosen real character, and assuming the generalised Riemann hypothesis for that $L$ -function. An example application of the method to one of the RSA challenge numbers will be given. Some optimisations of the method will be discussed. This is joint work with Andy Booker and Jon Keating.
10/07/2013 - Zhi Qi (OSU) - Hybrid subconvexity bound of $L(1/2, Sym^2 f \times g)$ (abstract)

The subconvexity problem of the central values of L-functions is outstanding in analytic number theory. For instance, that for $L(1/2, Sym^2 f \times \varphi)$ , with $f$ an automorphic cusp form and $\varphi$ a given Maass form, is closely related to the quantum unique ergodicity problem in view of Watson's formula. Let $f$ be a holomorphic form of full level and weight $k$ , and $g$ a holomorphic newform of prime level $p$ and fixed weight. With $k$ and $p$ large and varying under the condition $p^{13/64+\delta} < k < p^{3/8+\delta}$ , we prove a subconvexity bound for $L(1/2, Sym^2 f \times g)$ in the aspect of both $k$ and $p$ .
09/23/2013 - Fan Zhou (OSU) - Sato-Tate Equidistribution of Satake Parameters of Maass Forms on PGL(N) (abstract)

The equidistribution of Hecke eigenvalues of a family of automorphic forms on GL(2) has been studied by Serre, Sarnak, Bruggeman, Conrey-Duke-Farmer, etc. We formulate a conjectured orthogonality relation between the Fourier coefficients of Maass forms on PGL(N) for N>=2. Based on the work of Goldfeld-Kontorovich and Blomer for N=3, and on our conjecture for N>=4, we prove a weighted vertical equidistribution theorem (with respect to the generalized Sato-Tate measure) for the Satake parameter of Maass forms at a finite prime.

2012-2013

04/23/2013 - Guillaume Ricotta (Bordeaux) - Fourier coefficients of GL(3) automorphic forms in arithmetic progressions (abstract)

We prove that the Fourier coefficients of GL(3) Hecke-Maass cusp forms in arithmetic progression modulo a prime number p have a Gaussian limit distribution as p goes to infinity. This is a joint work with Emmanuel Kowalski.
04/08/2013 - Craig Franze (OSU) - An asymptotic expansion related to the Dickman function. (abstract)

In a recent paper Soundararajan proved a conjecture of Broadhurst, giving an asymptotic expansion for a sequence of integrals related to the Dickman function. In this talk, I will discuss a generalization of this expansion, as well as its implications to other number-theoretic functions arising as solutions to delay equations.
04/01/2013 - Joachim Schwermer (Vienna) - On Lefschetz numbers and arithmetically defined hyperbolic 3-manifolds. (abstract)

An orientable hyperbolic 3-manifold is isometric to the quotient of hyper- bolic 3-space H by a discrete torsion free subgroup of the group of orientation-preserving isometries of H. Among these manifolds, the ones originating from arithmetically defi ned groups form a family of special interest. Due to the underlying connections with number theory and the theory of automorphic forms, there is a fruitful interaction between geometric and arithmetic questions, methods and results. We intend to give an account of recent investigations in this area, in particular, of those pertaining to the cohomology of these hyperbolic 3-manifolds. This includes a recent result concerning the growth of Betti numbers for compact arithmetic hyperbolic 3-manifolds.
02/25/2013 - Bruce Berndt (UIUC) - Unpublished Manuscripts Published with Ramanujan's Lost Notebook. (abstract)

Published with Ramanujan's lost notebook are several partial manuscripts. Some evidently were intended to be portions of papers that he had published. Others are partial manuscripts of papers that were never completed. In this lecture, we discuss examples of both types. For the former, we offer speculation on why Ramanujan never included the results in his published papers. The manuscripts are over a broad range of topics, including classical analysis, analytic number theory, diophantine approximation, and elementary mathematics.
02/04/2013 - Youness Lamzouri (York U) - Large Character Sums. (abstract)

In 1932, Paley constructed an infinite family of quadratic characters whose character sums become exceptionally large. In this talk, I will discuss recent work (joint with Leo Goldmakher) in which we obtain analogous results for characters of any fixed order. Previously our bounds were only known under the assumption of the Generalized Riemann Hypothesis.
01/14/2013 - Dimitris Koukoulopoulos (Montreal) - When the sieve works. (abstract)

Let $S(x,P)$ be the number of integers up to $x$ that have no prime factors from the set of primes $P \subset\{p \le x\}$ . In general, a naive probabilistic heuristic suggests that $S(x,P) \approx x\cdot \prod_{p\in P} (1-1/p)$ . Sieve methods yield good upper and lower bounds, of this size, when $P$ is a subset of the primes in $\{p \le x^{1/2-\epsilon} \}$ , but they are inapplicable if $P$ contains lots of primes $>x^{1/2}$ . Now, for such $P$ , the size of $S(x,P)$ has been studied in only a few cases. In the case when $P= \{y < p \le x\}$ , which is known to be the most extreme one, we have that $S(x,P)\approx x/u^u$ , $u=\log x/\log y$ , much less that the expected $x/u$ . Other than that not much is known, but it is expected that, as soon as $P$ does not contains too many big primes, the probabilistic heuristic is accurate. In this talk, I will show that this expectation is indeed accurate: if $\sum_{y < p \le x,\, p \notin P} 1/p \gg1$ for some $y\ge x^{O(1)}$ , then $S(x;P)$ has the predicted size. This is joint work with Andrew Granville and Kaisa Matom\"aki.
12/03/2012 - Ghaith Hiary (Bristol) - Computing Dirichlet $L$ -functions. (abstract)

A fast method for computing Dirichlet functions to a power-full modulus is presented. The method achieves power-savings in the $t$ and $q$ aspects.
11/19/2012 - Kevin Ford (UIUC) - Sets $S$ of primes with $p$ in $S$ and $q|(p-1)$ implying $q$ in $S$ . (abstract)

Consider a set $S$ of primes such that if $p$ in $S$ and $q|(p-1)$ , then $q$ is in $S$ . We describe applications of such sets to Carmichael's conjecture and recent work of the speaker, Konyagin and Luca on groups with Perfect Order Subsets. We also descibe a new bound for the counting function of such sets: either $S$ contains all primes or $S$ is extremely thin; the number of primes in $S$ that are less than $x$ is $O(x^{1-c})$ for some $c>0$ .
10/22/2012 - Micah Milinovich (Ole Miss) - Simple zeros of L-functions of modular forms. (abstract)

Let $f$ be a primitive holomorphic cusp form of weight $k$ , level $q$ , and character $\chi$ , and let $L(s,f)$ be its associated $L$ -function. I will discuss how to prove quantitative estimates for the number of simple non-trivial zeros of $L(s,f)$ under the assumption of the generalized Riemann Hypothesis. Even assuming GRH, this seems to be the first method capable of proving that infinitely many primitive degree two $L$ -functions have an infinitude of simple non-trivial zeros. If there is time, I will discuss an ongoing project where the condition " $L(s,f)$ has infinitude of simple non-trivial zeros" is related to the non-vanishing of a certain average of Dirichlet twists of the derivative of $L(s,f)$ at the central point. In principle, this condition can be checked numerically for any particular choice of $f$ . This is joint work with Nathan Ng (Lethbridge).
10/01/2012 - Tim All (OSU) - On p-adic annihilators of real ideal classes. (abstract)

Let F be a real abelian field, p a rational prime unramified in F, and O the valuation integers of a topological closure of F via some fixed embedding into the algebraic closure of the p-adic rationals. Using the techniques of Euler systems originally discovered by Thaine, David Solomon has conjectured that an explicit element of the Galois group ring with coefficients in O annihilates the ideal class group of F tensored with O. This conjecture is the real analogy of a classical result of Stickelberger concerning totally imaginary fields. Recently, we have obtained a proof of a strengthened version of this conjecture with no assumptions on p. We will outline the proof, and discuss some consequences/generalizations.
09/24/2012 - David Goss (OSU) - From Carlitz's module to Euler's Gamma function. (abstract)

A recent preprint of Federico Pellarin shows how it is possible to use the Carlitz formalism to obtain Euler's Gamma function in a manner similar to that of the Anderson-Thakur function in characteristic p. We will discuss this approach in this seminar.
09/17/2012 - Roman Holowinsky (OSU) - Hybrid Subconvexity Bounds for Rankin-Selberg Convolutions. (abstract)

We will consider various moment averages over special values of Rankin-Selberg convolution $L$ -functions $L(\frac{1}{2}, \pi_1 \times \pi_2)$ . Each $\pi_j$ will contribute to the size of the associated analytic conductor and we shall take advantage of this fact in order to obtain hybrid subconvexity bounds for individual $L$ -values. Topics presented here will be a combination of ongoing joint projects with Nicolas Templier and Ritabrata Munshi.
09/10/2012 - Jim Cogdell (OSU) - The local Langlands correspondence for GL(n) and the symmetric and exterior square $\varepsilon$ -factors (abstract)

Artin introduced his non-abelian L-functions for representations of the Galois group in a series of papers in 1923--1931. He was able to define the local Euler factors for all primes and define the Artin conductor that appears in the functional equation, but the Artin root number remained mysterious. It was factored by Deligne in 1971 as part of his proof of the existence of the local $\varepsilon$ -factors that appear in the functional equation of the Artin L-functions. One way too understand these L-functions and $\varepsilon$ -factors is to find a corresponding analytic object, and automorphic form, whose L-function and $\varepsilon$ -factors match the arithmetic ones. This is the content of the local Langlands correspondence. This correspondence should be robust and preserve various parallel operations on the arithmetic and analytic sides, such as taking the exterior square or symmetric square. In collaboration with F. Shahidi and T-L. Tsai, we have recently showed that indeed the local $\varepsilon$ -factors that appear in the functional equations are preserved under these operations. The proof is an application of local/global techniques and the stability of these factors under highly ramified twists. In this talk I will attempt to explain a bit about these objects and the techniques we use in our proof.

2011-2012

05/21/2012 - Sun Kim (OSU) - Bessel function series and the circle and divisor problems. (abstract)

In approximately 1915, Ramanujan recorded without proofs two identities involving doubly infinite series of Bessel functions. The two identities are connected with the classical circle and divisor problems, respectively. For each of Ramanujan's identities, there are three possible interpretations for the double series. The first identity was proved under those three and the second identity was proved under two of them. In this talk, we briefly review some key aspects of the proofs. Furthermore, we discuss trigonometric analogues of the identities, and a generalization of the first identity in the setting of Riesz sums. This is joint work with Bruce Berndt and Alexandru Zaharescu.
04/30/2012 - Ben Brubaker (MIT) - Demazure operators and Iwahori Whittaker functions. (abstract)

Around 1980, Casselman and Shalika gave an elegant proof of a formula (conjectured by Langlands) for the spherical Whittaker function of an unramified principal series. We will begin by reviewing these terms and some key aspects of the proof, which relies on clever manipulations in the Iwahori-Hecke algebra. Then we explore similar formulas for Iwahori-fixed vectors in the unramified principal series, where we "rediscover" a Hecke algebra action due to Lusztig, in the context of equivariant K-theory. (This is joint work with Dan Bump and Tony Licata).
04/23/2012 - Hourong Qin (Nanjing) - The Local $L$ -series of CM Elliptic Curves and Quadratic Polynomials Represent Primes. (abstract)

Let $E$ be an elliptic curve defined over $\mathbb{Q}$ with complex multiplication. Fix an integer $r$ . We give sufficient and necessary conditions for $a_p=r$ for some prime $p$ . We show that there are infinitely many primes $p$ such that $a_p=r$ for some fixed integer $r$ if and only if a quadratic polynomial represents infinitely many primes $p$ . In particular, for $E: y^2=x^3+x$ , our result inplies that there are infinitely many primes $p$ such that $a_p=2$ if and only if there are infinitely many primes $p$ of the form $n^2+1$ .
04/16/2012 - Peng Zhao (Princeton) - The Quantum Variance of PSL(2,Z)\PSL(2,R). (abstract)

We discuss the quantum variance, which is introduced by Zelditch and describes the fluctuations of a quantum observable, on the phase space of modular surface. We asymptotically evaluate the quantum variance and show that it is equal to the classical variance of the geodesic flow on the phase space after inserting the "correction factor" of certain L-function's central value on each irreducible subspace. It is also very close to the arithmetic variance studied by Luo, Rudnick and Sarnak. This talk is based on a joint work with Peter Sarnak.
03/12/2012 - Henry Kim (Toronto) - Application of the strong Artin conjecture to the class number problem. (abstract)

As an application of the strong Artin conjecture, we exhibit a family of number fields unconditionally with extreme class numbers, whose normal closures have S5,S4,A4, and dihedral groups Dn,n=3,4,5, and cyclic groups Cn,n=4,5,6, as their Galois groups. This is a joint work with P.J. Cho.
02/09/2012 - Guillaume Ricotta (Bordeaux) - Height of Heegner points. (abstract)

The asymptotic height of Heegner points, on average over large discriminants, is investigated. In particular, the second order term is obtained. This is a joint work with Nicolas Templier.
12/05/2011 - Nathan Conrad Jones (University of Mississippi) - An alternative view of primitivity of Dirichlet characters. (abstract)

Dirichlet characters and their associated L-functions were introduced by Dirichlet in his proof of the prime number theorem in arithmetic progressions. Recall that a Dirichlet character is called imprimitive if it is induced from a character of smaller level, and otherwise it is called primitive. In this talk, I will discuss a modification of "inducing to higher level" which causes imprimitive characters to behave primitively (e.g. the properties of the associated Gauss sum and the functional equation of the attached L-function take on a form usually associated to a primitive character). This is based on joint work with R. Daileda.
10/18/2011 (1:30 pm) - Amy DeCelles (Goshen) - Number theoretic applications of the automorphic spectral theory of higher rank groups. (abstract)

Diaconu, Garrett, and Goldfeld have exhibited a general construction of spectral identities involving sums of integral moments for any Rankin-Selberg integral representation of L-functions. We construct a Poincare series, formed from an explicit solution to a differential equation on a complex symmetric space, suitable for producing such an identity for Rankin-Selberg convolutions for GL(n) x GL(n) over totally complex number fields. A sample application of this Poincare series is an exact formula relating the automorphic spectrum of the Laplacian to the number of lattice points in an expanding region in a complex symmetric space.
10/10/2011 - Federico Pellarin (St. Etienne) - On the values of certain L-series at ``even" positive integers. (abstract)

In this talk, we first briefly discuss some of the many existing ways to compute the values of classical zeta functions (e.g. Riemann's and Dedekind's) at even positive integers. Then we survey similar questions in the framework of the arithmetic of function fields in positive characteristic. In that framework, we propose a formula for the value at one for a family of L-series allowing to recover known results for values of Carlitz-Goss zeta functions and L-series associated to Dirichlet characters.
10/03/2011 - Kenneth Ward (Oklahoma State) - An asymptotic relation of class number and genus for abelian extensions of a function field. (abstract)

I demonstrate an asymptotic relation of class number and genus for the abelian extensions of a fixed choice of congruence function field. This result has been compared to the Brauer-Siegel theorem of number fields, but is not a precise analogue. This remains an open problem beyond the abelian case; I will explain why this is so.
09/19/2011 - Abhishek Saha (ETH) - Determination of modular forms by Fourier coefficients. (abstract)

I will describe my recent work, part of which is joint with Ralf Schmidt, that shows that a Siegel cusp form of degree 2 under certain assumptions is determined by its set of Fourier coefficients a(S) with -4 det(S) ranging over fundamental discriminants. As a key step to the result, I will prove a very similar fact for holomorphic cusp forms of half-integral weight that generalizes an old result of Luo-Ramakrishnan. I will also briefly describe some important consequences of the main result for the L-functions and Bessel models related to such a Siegel cusp form. In particular, an application to the case of Yoshida lifts leads to a simultaneous non-vanishing theorem for two Rankin-Selberg L-functions.

2010-2011

05/16/2011 - Sheng-Chi Liu (Texas A&M) - The L^2 restriction norms of Satio-Kurokawa lifts. (abstract)

A Siegel modular form, when restricted to a certain natural submanifold of Siegel's upper half space, is essentially a classical elliptic modular form in each of two variables. In the special case that the Siegel form is a Saito-Kurokawa lift, Ichino gave a formula which explicitly decomposes this restricted Siegel form into elliptic modular forms; the formula involves central values of Rankin-Selberg L-functions on GL3 x GL2. I will talk about some recent results on the average behavior of these L-functions which give some information on how the restricted Siegel form usually behaves. This is joint work with Matt Young.
05/09/2011 - Solomon Friedberg (Boston College) - Eisenstein series, crystals and ice. (abstract)

I describe how the Whittaker coefficients of Eisenstein series may be described in two new ways: using crystal graphs and using ice models from statistical mechanics. This is joint work with Ben Brubaker and Daniel Bump.
05/02/2011 - Ling Long (Iowa State) - Recent developments on modular forms for noncongruence subgroups. (abstract)

Among all finite index subgroups of the modular group, majority of them are noncongruence, i.e. they cannot be described in terms of congruence relations. The systematic investigation of modular forms for noncongruence subgroups was initiated by Atkin and Swinnerton-Dyer in 1960's. Compared to the classical theory of congruence modular forms, noncongruence modular forms are more mysterious due to the lack of efficient Hecke theory. However, noncongruence modular forms exhibit some remarkable properties and are closely related to many topics in number theory and beyond. In this talk, we will introduce these functions, discuss some recent developments in this area as well as applications and future research directions.
02/28/2011 - Jeff Hoffstein (Brown) - Shifted multiple Dirichlet series and moments of L-series. (abstract)

I'll explain what shifted convolutions are and show how they can interact with the theory of multiple Dirichlet series, with number theoretic applications. I will assume no previous knowledge of either subject.
02/21/2011 - Gautam Chinta (CUNY) - Orthogonal periods of Eisenstein series. (abstract)

We will show an identity between an orthogonal period of a minimal parabolic Eisenstein series on GL(3) and Whittaker coefficients of an Eisenstein series on the metaplectic double cover of GL(3), thereby providing evidence in favor of a conjecture of Jacquet. The main tool used in the proof is Gauss's three squares theorem. This is joint work with O. Offen.
12/03/2010 - Noam Elkies (Harvard) - On the areas of rational triangles. (abstract)

By a "rational triangle" we mean a plane triangle whose sides are rational numbers. By Heron's formula, there exists such a triangle of area a^(1/2) if and only if a>0 and xyz(x+y+z)=a for some rationals x,y,z. In a 1749 letter to Goldbach, Euler constructed infinitely many such (x,y,z) for any rational a (positive or not), remarking that it cost him much effort, but not explaining his method. We suggest one approach, using only tools available to Euler, that he might have taken, and use this approach to construct several other infinite families of solutions. We then reconsider the problem as a question in arithmetic geometry: xyz(x+y+z)=a gives a K3 surface, and each family of solutions is a singular rational curve on that surface defined over Q. The structure of the Neron-Severi group of that K3 surface explains why the problem is unusually hard. Along the way we also encounter the Niemeier lattices (the even unimodular lattices in R^24) and the non-Hamiltonian Petersen graph.
11/22/2010 - Aaron Levin (Michigan State) - Runge's effective method for integral points. (abstract)

Among the few known effective Diophantine techniques in number theory is an old method of Runge for effectively computing integral points on certain affine curves. I will review Runge's method, including some recent extensions and generalizations. I will then discuss various applications of Runge's method to specific curves and varieties of interest.
11/16/2010 - Lenny Taelman (Leiden) - The Carlitz sheaf, cyclotomic function fields, and Vandiver's conjecture. (abstract)

In this talk I will discuss conjectural function field analogues of the Herbrand-Ribet theorem and of Vandiver's conjecture. Also, I will sketch how these relate to previous work of David Goss and Warren Sinnott. Along the way I will explain the basics of Carlitz' theory of cyclotomic function fields in a more modern framework.
11/15/2010 - Xiaoqing Li (SUNY Buffalo) - The L^2 restriction of a GL(3) Maass form to GL(2). (abstract)

In this talk, we will study the L^2 restriction problem of a GL(3) Maass form to GL(2). By Parseval's formula, the problem becomes bounding averages of different families of GL(3)xGL(2) L-functions. Assuming the Lindelof hypothesis for these GL(3)xGL(2) L-function\ s as we usually do, one can achieve a sharp bound in terms of the analytic conductor of the varying GL(3) Maass form. However, we will give an unconditional proof of this sharp bound for selfdual GL(3) Maass forms. For nonselfdual GL(3) Maass forms, our bounds depend on the bounds of the first Fourier coefficients of the GL(3) Maass forms. This is joint work with Matthew Young.
11/08/2010 - Paul Nelson (Caltech) - Equidistribution of cusp forms in the level aspect. (abstract)

Let f traverse a sequence of classical holomorphic newforms of fixed weight and increasing squarefree level q tending to infinity. We prove that the pushforward of the mass of f to the modular curve of level 1 equidistributes with respect to the Poincare measure.
11/01/2010 - Eddie Herman (AIM) - Beyond Endoscopy for the Asai L-function and Quadratic Base Change. (abstract)

Using Langlands' "Beyond Endoscopy" idea and analytic number theory techniques, we study the Asai L-function associated to a real quadratic field K/Q. If the Asai L-function associated to an automorphic form over K has a pole, then the form is a base change from Q. We show how to prove this and the analytic continuation of the L-function. This is one of the first examples of using a trace formula to get such information. Time permitting, other ideas related to beyond endoscopy will be addressed.
10/25/2010 - David Goss (OSU) - The class number formula of Lenny Taelman. (abstract)

Characteristic $p$ arithmetic can be viewed as a laboratory that allows us to study classical objects "through the looking glass." In a sense this is similar to what one would expect from life based on silicon as opposed to carbon.

The motivating idea in characteristic $p$ arithmetic is precisely to use analysis in finite characteristic where in classical number theory one uses complex analysis of course. As classical arithmetic is based on the integers, characteristic $p$ arithmetic is based on Fq[t] (or even much more general base rings). Moreover, algebraic number theory is ties together such invariants as the class number with special values of $L$ -series.

In this talk we shall describe the trailblazing recent work of Lenny Taelman where Fq[t]-analogs of the class group and regulator are introduced and an analytic class number proved.
10/18/2010 - Jim Cogdell (OSU) - On Local L-functions. (abstract)

When we study automorphic L-functions via integral representations we usually concentrate on finding a family of integrals that have nice analytic properties and ar\ e Eulerian and then performing the unramified calculation to identify the L-function. Seldom do we explicitly compute the L-function at the ramified or archimedean \ places. However, to use the converse theorem to approach functoriality we need better knowledge of what happens at these places. In this lecture I would like to dis\ cuss two ideas in this direction: the use of (Bernstein-Zelevinsky) derivatives and ``exceptional'' poles. These work well for GL(n) constructions (Rankin-Selberg, \ exterior square, Asai, symmetric square) and there are some indications that similar ideas may work for classical groups.
10/11/2010 - Yangbo Ye (Iowa) - Resonance of automorphic forms for GL(2) and GL(3). (abstract)

Resonance is an important phenomenon which may occur between two vibration systems. Fixing one vibration system, one may change the second (testing) system to detect resonance frequencies of the first, and hence gain spectral information on the oscillation nature of the first vibration system. A classical example of this is the Fourier series expansion of a periodic function, which is actually the GL(1) theory. In this talk we will work out the GL(2) case following a result by Luo-Iwaniec-Sarnak. Then we will consider a Maass cusp form f for SL(3,Z) and compute averages of its Fourier coefficients A(m,n) twisted by exponential functions e(a n^b) of a fractional power. We will show that a main term occurs when b=1/3 and m a^3/27 is an integer. The existence of such a main term manifests the vibration and resonance behavior of these automorphic forms for GL(3).
10/04/2010 - Timothy All (OSU) - Mihailescu's Theorem (formerly Catalan's conjecture); an outline of the proof. (abstract)

In 1844, Eugene Catalan submitted the following conjecture to the Journal de Crelle, that no two consecutive integers, save 8 and 9, are perfect powers. One hundred and sixty years later, Preda Mihailescu published the first proof using methods from the theory of cyclotomic fields. We give an overview of these methods and show how they produce a proof of Catalan's conjecture.

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