
THEMATIC PROGRAMS 

August 4, 2024  
Automorphic Forms ProgramWorkshop on Automorphic Lfunctions

James Arthur  JungJo 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 LanglandsShelstad 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 Lfunction of a 2dimensional 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
Weiltype 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 nearminimal 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 RankinSelberg 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, PiatetskiShapiro, 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 padic
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 BuniakowskiSchinzel 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 Lfunctions of elliptic curves
Let $E$ be an elliptic curve over the rationals with Lfunction $L_E(s)$.
For any character $\chi$, let $L_E(s, \chi)$ be the twisted Lfunction.
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 Lfunctions
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 SwinnertonDyer.
This is a joint work with Ram Murty.
JungJo 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 Lfunctions
In this talk we shall describe joint work with Ram Murty on formulating a pair correlation conjecture in the context of Artin Lfunctions. 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 nonArchimedean local field whose residue characteristic
is odd. These are analogous to results of Casselman for ${\rm GL}_2(F)$
and Jacquet, PiatetskiShapiro 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 KimShahidi Transfer of $GL (2)\times
GL (3)$ to $GL (6)$
The talk will be mainly based on my recent paper "On Symmetric
5th powers of Cusp Forms on $GL(2)$ of icosahedral type." It was
motivated by a Banff conference talk given by Kim.