Seminars this semester

Feb 13	Tue	Luis Santiago Palacios (Bordeaux)	Number Theory seminar
13:00		Geometry of the Bianchi eigenvariety at non-cuspidal points
		Hicks Seminar Room J11 / Google Meet
	Abstract: An important tool to study automorphic representations in the framework of the Langlands program, is to produce $p$-adic variation. Such variation is captured geometrically in the study of certain "moduli spaces" of p-adic automorphic forms, called eigenvarieties. In this talk, we first introduce Bianchi modular forms, that is, automorphic forms for $\mathrm{GL}_2$ over an imaginary quadratic field, and then discuss its contribution to the cohomology of the Bianchi threefold. After that, we present the Bianchi eigenvariety and state our result about its geometry at a special non-cuspidal point. This is a joint work in progress with Daniel Barrera (Universidad de Santiago de Chile).
Feb 20	Tue	Beth Romano (Kings College London)	Number Theory seminar
13:00		Epipelagic representations in the local Langlands correspondence
		Hicks Seminar Room J11 / Google Meet
	Abstract: The local Langlands correspondence (LLC) is a kaleidoscope of conjectures relating local Galois theory, complex Lie theory, and representations of p-adic groups. The LLC is divided into two parts: first, there is the tame or depth-zero part, where much is known and proofs tend to be uniform for all residue characteristics p. Then there is the positive-depth (or wild) part of the correspondence, where there is much that still needs to be explored. I will talk about recent results that build our understanding of this wild part of the LLC via epipelagic representations and their Langlands parameters. I will not assume background knowledge of the LLC, but will give an introduction to these ideas via examples.
Feb 27	Tue	Alexandros Groutides (Warwick)	Number Theory seminar
13:00		On integral structures in smooth $\mathrm{GL}_2$-representations and zeta integrals.
		Hicks Seminar Room J11 / Google Meet
	Abstract: We will discuss recent work on local integral structures in smooth ($\mathrm{GL}_2\times H$)-representations, where $H$ is an unramified maximal torus of $\mathrm{GL}_2$. Inspired by work of Loeffler-Skinner-Zerbes, we will introduce certain unramified Hecke modules containing lattices with deep integral properties. We'll see how this approach recovers a Gross-Prasad type multiplicity one result in this unramified setting and present an integral variant of it with applications to zeta integrals and automorphic modular forms. Finally, we will reformulate and answer a conjecture of Loeffler on integral unramified Hecke operators attached to the lattices mentioned above.
Mar 5	Tue	Lewis M Combes	Number Theory seminar
13:00		Period polynomials of level 1 Bianchi modular forms
		Hicks Seminar Room J11 / Google Meet
	Abstract: The period polynomial of a classical modular form encodes important arithmetic information about the form itself, being made out of critical L-values and connecting to congruences via Haberland's formula. In this talk, we report on work to generalise these connections to the setting of Bianchi modular forms---those over an imaginary quadratic field. We demonstrate explicit congruences between various types of Bianchi modular form, and show how to detect them using a pairing on period polynomials.
Mar 12	Tue	Andrea Dotto (Cambridge)	Number Theory seminar
13:00		Some consequences of mod p multiplicity one for Shimura curves
		Hicks Seminar Room J11 / Google Meet
	Abstract: The multiplicity of Hecke eigenspaces in the mod p cohomology of Shimura curves is a classical invariant, which has been computed in significant generality when the group is split at p. This talk will focus on the complementary case of nonsplit quaternion algebras, and will describe a new multiplicity one result, as well as some of its consequences regarding the structure of completed cohomology. I will also discuss applications towards the categorical mod p Langlands correspondence for the nonsplit inner form of GL_2(Q_p). Part of the talk will comprise a joint work in progress with Bao Le Hung.
Apr 23	Tue	Johannes Droschl	Number Theory seminar
13:00		On modular representations of $GL_n$ over a p-adic field
		Hicks Seminar Room J11 / Google Meet
	Abstract: The Godement-Jacquet L-function is a classical invariant attached to irreducible representations of $GL_n$. Minguez extended their definition to representations over fields of characteristic $\ell\neq p$. In this talk we will finish the computation of these L-functions for modular representations and check that they agree with the L-function of their respective C-parameter defined by Kurinczuk and Matringe. We approach the problem by extending the theory of square-irreducible representations, and their derivatives, of Lapid and Minguez to modular representations and applying it to our setting.
Apr 30	Tue	Bence Hevesi (Kings College London)	Number Theory seminar
13:00		Local-global compatibility at l=p for torsion automorphic Galois representations
		Hicks Seminar Room J11 / Google Meet
	Abstract: Some ten years ago, Scholze proved the existence of Galois representations associated with torsion eigenclasses appearing in the cohomology of locally symmetric spaces for GL_n over imaginary CM fields. Since then, the question of local-global compatibility for these automorphic Galois representations has been an active area of research motivated by applications towards new automorphy lifting theorems. I will report on my work on local-global compatibility at l=p in this direction, generalising the results of the celebrated 10-author paper and Caraiani—Newton.
May 7	Tue	Jay Taylor (Manchester)	Number Theory seminar
13:00
		Hicks Seminar Room J11 / Google Meet
May 21	Tue	Owen Patashnick (Kings College London)	Number Theory seminar
13:00
		Hicks Seminar Room J11 / Google Meet