# THEMATIC PROGRAMS

April 25, 2017

## Workshop on Automorphic L-functions May 5-9, 2003 Organizers: H. Kim and R. Murty

 James Arthur Jung-Jo Lee Andrew Booker Philippe Michel William Casselman Kumar Murty Jim Cogdell Yiannis Petridis Alina Carmen Cojocaru A. Raghuram Chantal David Song Wang Wentang Kuo

James Arthur, University of Toronto
Weighted orbital integrals and transfer

Weighted orbital integrals arise naturally in harmonic analysis. As the geometric terms in the global trace formula, they play an important role in the study of harmonic analysis. One has to understand the behaviour of these terms under Langlands-Shelstad transfer in order to compare trace formulas on different groups.

We shall begin with an elementary introduction to weighted orbital integrals, while attempting to show that they are indeed natural objects.
We shall then describe some of the problems of transfer, and how these problems can be solved.

Andrew Booker, Princeton University
Converse theorems and Artin's conjecture

I will show that the L-function of a 2-dimensional complex Galois representation is either modular or has infinitely many poles. In particular, Artin's conjecture for a given representation implies modularity, with no twists needed. The technique used in the proof may be extended to give a new Weil-type converse theorem.

Daniel Bump, Stanford University
Small representations for odd orthogonal groups

Odd orthogonal groups beyond SO(7) do not have minimal representations. However they do have representations attached
to a near-minimal coadjoint orbit. These are representations of a metaplectic cover. It will be shown that they are automorphic and square integrable and have many vanishing Fourier coefficients which allow them to be used as theta kernels and potentially in the Rankin-Selberg method. Joint work with Friedberg and Ginzburg.

William Casselman, University of British Columbia
The Plancherel measure for Eisenstein series in higher rank

In a recent paper I showed how to apply elementary Fourier analysis and truncation on arithmetic quotients to derive the Plancherel measure for Eisenstein series in that case. With a little luck I'll explain here how to extend that derivation to quotients of higher rank.

Jim Cogdell, Oklahoma State University/The Fields Institute
Functoriality for the classical groups

Recently, in collaboration with Kim, Piatetski-Shapiro, and Shahidi, we have established global functoriality from the split orthogonal groups and symplectic groups to GL(N). Supposedly, one of the consequences of functoriality is the ability to then pull back'' various structural facts about automorphic representations of GL(N) to these groups. In this lecture I would like to explain how functoriality combined with the descent theory of Ginzburg, Rallis, and Soudry let us carry this out and obtain such global results as Ramanujan type bounds and rigidity theorems for globally generic cuspidal representations as well local results such as extending local functoriality and the local Langlands conjecture to all generic representations of SO(2n+1) over a p-adic field.

Alina Carmen Cojocaru, The Fields Institute
Applications of the Chebotarev density theorem to elliptic curves

Many classical questions from the theory of numbers can be formulated in the context of elliptic curves. Among them, Artin's primitive root conjecture, the twin prime conjecture, the Buniakowski-Schinzel hypothesis. We will show how to use effective versions of the Chebotarev density theorem and sieve methods to investigate elliptic curve analogues of these particular questions.

Chantal David, Concordia University
On the vanishing of twisted L-functions of elliptic curves

Let $E$ be an elliptic curve over the rationals with L-function $L_E(s)$. For any character $\chi$, let $L_E(s, \chi)$ be the twisted L-function. We show in this talk how Random Matrix Theory can be used to predict vanishing of L_E(s, \chi) in the family of cubic twists.

(joint work with J. Fearnley and H. Kisilevsky)

Wentang Kuo, Queen's University
Selberg's conjectures and L-functions

We will discuss the partial sums of coefficients of the Selberg class and apply this to study partial sums for the original conjecture by Birch and Swinnerton-Dyer.

This is a joint work with Ram Murty.

Jung-Jo Lee, Queen's University
An application of Mumford's gap principle

Let $E$ be an elliptic curve defined over $\bQ$. Its quadratic twists are denoted by $E_D$. In 1960, Honda made a surprising conjecture that $\rank_\bZ E_D(\bQ)$ is bounded as $D$ varies over all integers. At present, there is no evidence for or against this
conjecture.

Recently, Rubin and Silverberg derived an equivalent formulation of Honda's conjecture. Given an elliptic curve $E$, they construct certain infinite series related to $E$. They show that the ranks of elliptic curves in a family of quadratic twists are bounded if and only if these series converge.

Their theorem suggests the study of the cognate series for hyperelliptic curves. In this case, the convergence is an immediate consequence of a conjecture of Caporaso, Harris and Mazur. We prove this unconditionally applying Mumford's gap principle. This is a joint work with Ram Murty.

Philippe Michel, Universite Montpellier 11
On the shifted convolution problem

The Shifted Convolution Problem (SCP) is a seemingly technical, yet natural, problem going back to Kloosterman and Ingham. It aims at showing cancellations in partial sums of Rankin/Selberg type where one argument has a non trivial additive shift. One application of the SCP is to provide full solution of many instances of another problem: the Subconvexity Problem for $L$-functions (ScP). It this talks we review the various methods and ingredients that can be put into the solution of the SCP as well a few new instances of the ScP one can get with it.

Kumar Murty, University of Toronto
Pair Correlation and Artin L-functions

In this talk we shall describe joint work with Ram Murty on formulating a pair correlation conjecture in the context of Artin L-functions. In particular, we shall discuss the implications of such a conjecture for the Chebotarev density theorem.

Yiannis Petridis, The City University of New York
Distribution of modular symbols: (joint with M. S. Risager)

The modular symbols are defined as $\langle \gamma, f\rangle=-2\pi i \int_{a}^{\gamma a }f(\tau )\d\tau,$ where $f(\tau )$ is a holomorphic cusp form of weight $2$ for $\Gamma$ and $\gamma\in \Gamma$. We prove that on a surface with cusps the modular symbols appropriately normalized and ordered according to $c^2+d^2$, where $(c, d)$ is the second row of $\gamma$ have a binormal
distribution in the complex plane with correlation coefficient $0$. We examine various possible generalisations.

A. Raghuram, Purdue University
Conductors and Newforms for SL(2)

In this talk I will present some recent results obtained in collaboration with Joshua Lansky. This is about a theory of newforms for ${\rm SL}_2(F)$ where $F$ is a non-Archimedean local field whose residue characteristic is odd. These are analogous to results of Casselman for ${\rm GL}_2(F)$ and Jacquet, Piatetski-Shapiro and Shalika for ${\rm GL}_n(F)$.

To a representation $\pi$ of ${\rm SL}_2(F)$ we attach an integer $c(\pi)$ that we call the conductor of $\pi$. The conductor of $\pi$ depends only on the $L$-packet $\Pi$ containing $\pi$. It is turns out to be equal to the conductor of a minimal representation of ${\rm GL}_2(F)$ determining the $L$-packet $\Pi$. We use the results for ${\rm SL}_2(F)$ to prove similar results for ${\rm U}(1,1)$, the quasi split unramified unitary group in two variables. A newform is a vector in $\pi$ which is essentially fixed by a congruence subgroup of level $c(\pi)$. For both the groups we show that our newforms are always test vectors for some standard Whittaker functionals and in doing so we give various explicit formulae for newforms and further it is used to formulate a multiplicity one theorem for these newforms.

Song Wang, Yale
On the cuspidality of the Kim--Shahidi Transfer of $GL (2)\times GL (3)$ to $GL (6)$

The talk will be mainly based on my recent paper "On Symmetric 5-th powers of Cusp Forms on $GL(2)$ of icosahedral type." It was motivated by a Banff conference talk given by Kim.