The aim of this seminar is to cover material not normally covered in graduate courses and related to Analytic Number Theory and Automorphic Forms.
This semester the seminar meets on Fridays at 3:30 PM in Room 4214-03 (Mathematics Thesis Room). The organisers are Yiannis Petridis (petridis at member.ams.org) and Gautam Chinta (chinta at sci.ccny.cuny.edu). For information on the seminar in previous semesters go to Collaborative Satellite Number Theory Seminar Fall 2006 Collaborative Satellite Number Theory Seminar Spring 2006
Title: Some Recent Instances and Applications of Functoriality
Abstract: In these talks we discuss a number of recent results in number theory and group representations which are all consequences of certain cases of Langlands functoriality conjecture. Among the application we mention upper bounds for sums of Kloosterman sums over an arithmetic progression (communications with Sarnak), holomorphy of the spin L-function for GSp(4) (joint with Asgari), existence of Siegel modular froms of weight 3 and higher and genus 2 which are neither of Saito-Kurokawa nor Yoshida types (joint with Ramakrishnan), orthogonal and symplectic representations of GL(n) and reducibility questions for classical groups (joint with Cogdell, Kim and Piatetski-Shapiro), as well as a brief mention of the work of Khare-Larsen-Savin on the inverse Galois problem. One of the main ingredients in most of these results is the functorial transfer of generic cuspidal representations of classical groups to GL(n) which we will discuss here in more generality now that the main obstacle in establishing them, i.e., stability of root numbers, is resolved for quasisplit classical groups in a joint paper with Cogdell and Piatetski-Shapiro.
Professor Shahidi will give two talks: 1:00-2:00 PM and 3:30-5:00 PM in Room 4419
Title: On spectrum and arithmetic
Title: Tame Supercuspidal Representations
Supercuspidal representations are the basic building blocks in the representation theory of reductive groups over p-adic fields. Relatively recently, J.-K. Yu established a general construction that associates supercuspidal representations to certain parametrizing data. This talk discusses the problem of determining when two different parameters yield equivalent representations. This is an unexpected consequence of general results involving symmetric spaces over p-adic fields obtained jointly with Fiona Murnaghan.
Title: Isospectral tori and automorphic forms: An introduction
When do the vectors in two lattices in R^n have same lengths, while the lattices are not the same? The first examples of Witt (n=16) give also the first examples of isospectral tori (Milnor) and were the first examples of isospectral manifolds. There is a complete theory in which dimensions the phenomena happens. Quite often we can prove that the lengths are the same by creating a theta series, which is a modular form and look at the dimension of the appropriate space of modular forms.
Title: A Sieve Method for Quantum Unique Ergodicity
We'll look at the problem of Quantum Unique Ergodicity on the classical quotient space SL(2,Z)\H by reducing to the study of shifted sums of Hecke eigenvalues of Maass cusp forms. We'll then analyze the average size of the shifted summation terms by means of an upper-bound sieve and obtain a non-trivial result which suggests cancellation among the terms may not be required for the proof of Quantum Unique Ergodicity. Generalizations can then be made for any shifted sum of multiplicative functions, including sums with multiple shifts.
Title: "Newforms for SL(2) and U(1,1)."
Abstract: The theory of newforms, originally developed by Atkin and Lehner in the classical context of cusp forms on the upper half-plane, was reinterpreted in terms of the representation theory of GL(2) by Casselman. The theory was later extended to GL(n) by Jacquet, Piatetski-Shapiro, and Shalika. The significance of newforms to the theory of automorphic forms will be discussed in this setting. We will then present some recent extensions of this theory to SL(2) and the quasi-split unramified unitary group U(1,1).
Title: Undecidable problems about rational points and conjectures about elliptic curves
Abstract. This talk is about joint work with Karim Zahidi (Antwerp/Ghent), in which we describe a conjecture about elliptic curves (existence of primitive divisors that are inert in a quadratic field) that implies the existence of an easy undecidable problem about rational points on varieties. After a short historical introduction to undecidability, I will provide some heuristics for the conjecture, and prove a weaker-in-density version.
Title: A p-adic construction of global points on elliptic curves over imaginary quadratic field
Abstract: An elliptic curve E over an imaginary quadratic field F is in most cases conjectured to correspond to a weight 2 cusp form on GL_2(A_F) (a la Shimura-Taniyama). Such forms admit an elementary description as harmonic differentials on quotients of the upper half-space, and are the only type of modular form other than on GL_2(Q) to posess an analogous, 1-dimensional modular symbol. Using this modular symbol we construct certain measures on P^1(C_p), p a characteristic of bad reduction, and define period integrals whose image under the Tate parametrization conjecturally yields points defined over class fields of a suitable quadratic extension K/F, thus giving an answer to Hilbert's 12th problem for some totally complex quartic fields K. The talk will have two parts: 1. We will present the numerical evidence for this construction, 2. In the case where E is defined over Q, we prove that the period of the corresponding modular form over F is, up to an explicit constant, a rational multiple of the area of E(C).
Title: Automorphic forms and the cohomology of arithmetic groups
The cohomology of arithmetic groups provides a concrete realization of certain kinds of automorphic forms, and thus provides one approach to explicitly study them. In this talk we explain this connection and describe some techniques to study these cohomology spaces. Our goal is to show in some cases how to compute the action of the Hecke operators on the "interesting" part of the cohomology, namely the part that contains the contributions of the cusp forms.
Title: Deformations of Maass forms
Abstract: The Laplacian on a non-compact surface does not, in general, have a discrete spectrum. An exception is the arithmetic surfaces which arise in number theory. I will discuss the behavior of the discrete spectrum in response to deformations an arithmetic surface, in particular numerical experiments and their relation to theoretical results.
Title: Whittaker functions, Eisenstein series, and finite order
Recently, it has been shown that Eisenstein series attached to smooth sections (as opposed to K-finite) are meromorphic. However, very easy examples have been given that show that other analytic properties (specifically, meromorphy of finite order) do not hold as in the K-finite case. This parallels the same situation for Whittaker functions. Some ideas within this framework have had applications in the automorphic context. In this talk, I will sketch some proofs of these results and constructions. I will first give a chronological motivation for looking at these problems, and will end with a related interesting open question.
Title: Unitary periods and the Jacquet's relative trace formula
Abstract: I will survey some of the recent developments in the study of unitary period integrals of automorphic forms on GL(n) over a quadratic extension. The main tool to study such periods is the relative trace formula. I will explain the relevant identity of distributions and how it is applied in order to obtain information on the automorphic spectrum. In particular, I will explain Jacquet's characterization of the image of quadratic base change in terms of non vanishing of unitary periods and present a joint formula with Erez Lapid for the anisotropic unitary period of certain cusp forms in terms of special values of L-functions.
Title: Traces of CM values of modular functions and weight 3/2 modular forms
Abstract: The values of the famous j-invariant at quadratic irrationalities in the upper half plane are known as singular moduli and are of particular interest in number theory. Zagier realized the generating series of the traces of the singular moduli as a classical meromorphic modular form of weight 3/2. In this talk, we will discuss this and related results and give a generalization to modular functions on Riemann surfaces of arbitrary genus. Furthermore, we realize a certain generating series of arithmetic intersection numbers and Faltings heights as the derivative of Zagier's Eisenstein series of weight $3/2$. This recovers a result of Kudla, Rapoport and Yang. This is joint work with Jan Bruinier.
SPECIAL TALK ON THURSDAY AT 4:30 PM
Permutation groups and irrationality measures.
Abstract: In 1996 G. Rhin and C. Viola introduced a new algebraic method to obtain qualitative and quantitative irrationality results for values of special functions. Their method is based upon the study of the structure of certain permutation groups arising from the action of some birational transformations on double or triple Euler-type integrals, combined with the Euler integral representation of the hypergeometric function.