Tag - Automorphic forms

Jingwei Xiao: A Unitary Analogue of Friedberg-Jacquet Periods and Central Values of Standard L Functions on GL(2n)

3rd December, 2020

Let G be a reductive group over a number field F and H a subgroup. Automorphic periods study the integrals of cuspidal automorphic forms on G over H(F)\H(AF). They are often related to special values of certain L-functions. One of the most notable cases is when (G,H)=(U(n+1)☓U(n), U(n)), and these periods are related to central values of Rankin-Selberg L-functions on GL(n+1)☓GL(n). In this talk, I will explain my work in progress with Wei Zhang that studies central values of standard L-functions on GL(2n) using (G,H)=(U(2n), U(n)☓U(n)) and some variants. I shall explain the conjecture and a relative trace formula approach to study it. We prove the required fundamental lemma using a limit of the Jacquet-Rallis fundamental lemma and Hironaka’s characterization of spherical functions on the space of non-degenerate Hermitian matrices. Also, the question admits an arithmetic analogue.

Shai Evra: Ramanujan Conjecture and the Density Hypothesis

19th November, 2020

The Generalized Ramanujan Conjecture (GRC) for GLn is a central open problem in modern number theory. Its resolution is known to yield several important applications. For instance, the Ramanujan-Petersson conjecture for GL2, proven by Deligne, was a key ingredient in the work of Lubotzky-Phillips-Sarnak on Ramanujan graphs.

One can also state analogues of (Naive) Ramanujan Conjectures (NRC) for other reductive groups. However, in the 70s Kurokawa and Howe-Piatetski-Shapiro proved that the (NRC) fails even for quasi-split classical groups.

In the 90s Sarnak-Xue put forth a Density Hypothesis version of the (NRC), which serves as a replacement of the (NRC) in applications.

In this talk I will describe a possible approach to proving the Density Hypothesis for definite classical groups, by invoking deep and recent results coming from the Langlands programme: The endoscopic classification of automorphic representations of classical groups due to Arthur, and the proof of the Generalized Ramanujan-Petersson Conjecture.

Tasho Kaletha: An explicit supercuspidal local Langlands correspondence

29th October, 2020

We will give an explicit construction and description of a supercuspidal local Langlands correspondence for any p-adic group G that splits over a tame extension, provided p does not divide the order of the Weyl group. This construction matches any discrete Langlands parameters with trivial monodromy to an L-packet consisting of supercuspidal representations, and describes the internal structure of these L-packets.

The construction has two parts. The depth-zero part involves generalizing to disconnected groups results of Lusztig on the decomposition of a non-singular Deligne-Lusztig induction. Higher multiplicities occur in this decomposition and are handled using work of Bonnafé-Dat-Rouquier. The positive-depth part involves functorial transfer from a twisted Levi subgroup, which is made possible by an improvement of Yu's construction of supercuspidal representations obtained in recent joint work with Fintzen and Spice, and consideration of Harish Chandra characters.

We will also discuss ongoing work towards related conjectures: Shahidi's generic L-packet conjecture, Hiraga-Ichino-Ikeda formal degree conjecture,  stability and endoscopic transfer.

Jessica Fintzen: Representations of p-adic groups and applications

8th October, 2020

The Langlands programme is a far-reaching collection of conjectures that relate different areas of mathematics including number theory and representation theory. A fundamental problem on the representation theory side of the Langlands program is the construction of all (irreducible, smooth, complex) representations of p-adic groups. I will provide an overview of our understanding of the representations of p-adic groups, with an emphasis on recent progress. I will also outline how new results about the representation theory of p-adic groups can be used to obtain congruences between arbitrary automorphic forms and automorphic forms which are supercuspidal at p, which is joint work with Sug Woo Shin. This simplifies earlier constructions of attaching Galois representations to automorphic representations, i.e. the global Langlands correspondence, for general linear groups. Moreover, our results apply to general p-adic groups and have therefore the potential to become widely applicable beyond the case of the general linear group.

James Maynard: Primes in arithmetic progressions to large moduli

2nd July, 2020

How many primes are there which are less than x and congruent to a modulo q? This is one of the most important questions in analytic number theory, but also one of the hardest - our current knowledge is limited, and any direct improvements require solving exceptionally difficult questions to do with exceptional zeros and the Generalized Riemann Hypothesis! If we ask for 'averaged' results then we can do better, and powerful work of Bombieri and Vinogradov gives good answers for q less than the square-root of x. For many applications this is as good as the Generalized Riemann Hypothesis itself! Going beyond this 'square-root' barrier is a notorious problem which has been achieved only in special situations, perhaps most notably this was the key component in the work of Zhang on bounded gaps between primes. I'll talk about recent work going beyond this barrier in some new situations. This relies on fun connections between algebraic geometry, spectral theory of automorphic forms, Fourier analysis and classical prime number theory. The talk is intended for a general audience.

Jayce Robert Getz: On triple product L-functions

7th May, 2020

Establishing the conjectured analytic properties of triple product L-functions is a crucial case of Langlands functoriality.  However, little is known.  I will present work in progress on the case of triples of automorphic representations on GL3; in some sense this is the smallest case that appears out of reach via standard techniques.  The approach is based on a the beautiful fibration method of Braverman and Kazhdan for constructing Schwartz spaces and proving analogues of the Poisson summation formula.

Henrik Gustafsson: Eulerianity of Fourier coefficients of automorphic forms

30th April, 2020

The factorization of Fourier coefficients of automorphic forms plays an important role in a wide range of topics, from the study of L-functions to the interpretation of scattering amplitudes in string theory.

In this talk I will present a transfer theorem which derives the Eulerianity of certain Fourier coefficients from that of another coefficient. I will also discuss some applications of this theorem to Fourier coefficients of automorphic forms in minimal and next-to-minimal representations.

Based on recent work with Dmitry Gourevitch, Axel Kleinschmidt, Daniel Persson and Siddhartha Sahi.

Paul Nelson: Eisenstein series and the cubic moment for PGL2

28th January, 2020

We will discuss how to study the cubic moment of any family of automorphic L-functions on PGL2 using regularized diagonal periods of Eisenstein series, following a strategy suggested by Michel-Venkatesh. Applications include generalizations to the setting of number fields of some results of Conrey-Iwaniec and Petrow-Young, improved estimates for representation numbers of ternary quadratic forms over number fields, and improvements to the prime geodesic theorem on arithmetic hyperbolic 3-folds.

Raphael Steiner: Taking the Hecke algebra to its limits

12th September, 2019

We parametrize elements in the full Hecke algebra in a way such that the parametrization represents a generic automorphic form. By convolving, we then arrive at pre-trace formulas which are modular in three variables. From here, various identities for higher moments may be derived. We give applications to the sup-norm and fourth-norm of holomorphic Hecke eigenforms as well as Hecke-Maass forms on Γ \ ℍ and furthermore outline future work on higher moments of periods and quantum variance. This is joint work with Ilya Khayutin.

Yiannis Sakellaridis: Beyond Endoscopy: Local aspects of Venkatesh’s thesis

14th March, 2019

The thesis of Akshay Venkatesh obtains a "Beyond Endoscopy" proof of stable functorial transfer from tori to SL2, by means of the Kuznetsov formula. In this talk, I will show that there is a local statement that underlies this work; namely, there is a local transfer operator taking orbital measures for the Kuznetsov formula to test measures on the torus. The global comparison of trace formulas is then obtained as a Poisson summation formula for this transfer operator.