Tag - Langlands programme

Linus Hamann: Some Progress on Categorical Local Langlands

30th March, 2022

Recently, Fargues and Scholze attached a semi-simple L-parameter to any smooth irreducible representation of a p-adic reductive group, realizing the local Langlands correspondence as a geometric Langlands correspondence over the Fargues-Fontaine curve. They conjectured that there should exist an analogue of the geometric Langlands conjecture in this setting, known as the categorical local Langlands correspondence. Concretely, this conjecture translates to the belief that certain Shtuka spaces, generalizing the Lubin-Tate and Drinfeld towers appearing in the work of Harris-Taylor, should have cohomology dictated by the semi-simple L-parameter that they construct. In this talk, we will explain how one can make some progress on this conjecture by showing the Fargues-Scholze correspondence is compatible with known instances of the local Langlands correspondence through global methods, and then using this compatibility together with techniques from geometric Langlands to fully describe the cohomology of these Shtuka spaces in certain cases.

Victor Wang: Conditional approaches to sums of cubes

18th November, 2021

In 1986, Hooley applied (what practically amounts to) the general Langlands reciprocity (modularity) conjecture and GRH in a fresh new way, over certain families of cubic 3-folds. This eventually led to conditional near-optimal bounds for the number of integral solutions to x13+...+x63 in expanding boxes.

Building on Hooley's work, I will sketch new applications of large-sieve hypotheses, the Square-free Sieve Conjecture, and predictions of Random Matrix Theory type, over the same geometric families - e.g. conditional statistical results on sums of three integer cubes (a project suggested by Amit Ghosh and Peter Sarnak). These form the bulk of my thesis work (advised by Sarnak), and involve phenomena both random and structured, average- and worst-case, and multiplicative and additive.

Paul Nelson: Bounds for standard L-functions

7th October, 2021

We consider the standard L-function attached to a cuspidal automorphic representation of a general linear group. We present a proof of a subconvex bound in the t-aspect. More generally, we address the spectral aspect in the case of uniform parameter growth.

Gal Dor: Monoidal Structures on GL2-Modules and Abstractly Automorphic Representations

4th March, 2021

Consider the function field F of a smooth curve over 𝔽q, with q>2. L-functions of automorphic representations of GL2 over F are important objects for studying the arithmetic properties of the field F. Unfortunately, they can be defined in two different ways: one by Godement-Jacquet, and one by Jacquet-Langlands. Classically, one shows that the resulting L-functions coincide using a complicated computation. Each of these L-functions is the GCD of a family of zeta integrals associated to test data. I will categorify the question, by showing that there is a correspondence between the two families of zeta integrals, instead of just their L-functions. The resulting comparison of test data will induce an exotic symmetric monoidal structure on the category of representations of GL2. It turns out that an appropriate space of automorphic functions is a commutative algebra with respect to this symmetric monoidal structure. I will outline this construction, and show how it can be used to construct a category of automorphic representations.

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.

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.