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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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 $c>0$.

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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).

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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.

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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.

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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.

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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.

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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.

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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).

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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 $n^2+1$.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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).

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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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