London Number Theory Seminar

UCL King's Imperial College

The London Number Theory Seminar is held weekly, on Wednesdays, during term time. The location of the seminar cycles between KCL, Imperial College and UCL. Here is our diversity policy.

This term (Autumn 2019), the seminar will be hosted by Imperial College, and will be held on most Wednesdays in Huxley room 340, starting at 1600. The exception is Wednesday, October 16, when the seminar will be held in Huxley room 213 - the Clore Lecture theatre. The organisers are Ana Caraiani, Robert Kurinczuk, and Matteo Tamiozzo. The talks start on Wed 9th October and finish on 11th December. The talks will be preceded by tea at 1530 in the common room on the 5th floor of Huxley. 2 October 2019 - No seminar because of Clay research conference in Oxford

9 October 2019 - Johannes Nicaise (Imperial College)
Title: Convergence of p-adic measures to Berkovich skeleta
Abstract: This talk is based on joint work with Mattias Jonsson (Michigan). The theory of mirror symmetry predicts that the fibers of a maximally unipotent degeneration of polarized complex Calabi-Yau n-folds converge to an n-sphere with respect to the Gromov-Hausdorff metric. Boucksom and Jonsson have shown that, if we choose a family of volume forms on these Calabi-Yau manifolds, then the induced measures converge to a Lebesgue measure on Kontsevich and Soibelman’s essential skeleton of the degeneration, which conjecturally coincides with the Gromov-Hausdorff limit. This convergence takes place in a suitable Berkovich space that contains both the complex fibers and the non-archimedean nearby fiber of the degeneration. In this talk, I will explain a $p$-adic version of this result, answering a question that was raised by Matt Baker.

Friday October 11th, 1-2pm in Huxley 642 One-off seminar at non-standard time and date and place:
João Lourenço (Bonn) - "Integral affine Graßmannians of twisted groups and local models of Shimura varieties".
Abstract: Local models of Shimura varieties are integral models of flag varieties which help in understanding the local geometric behaviour of arithmetic models of Shimura varieties and were first systematically introduced by Rapoport-Zink in EL and PEL cases. More recently, a group-theoretic approach to their definition and study has been possibilitated by the theory of affine Graßmannians, as in the works of Pappas-Rapoport and Pappas-Zhu, where the authors always assume tame ramification. We generalise the constructions of these last papers, by exhibiting certain smooth affine and connected "parahoric" group models over Z[t] of a given quasi-split Q(t)-group G with absolutely simple simply connected cover splitting over the normal closure of Q(t^{1/e}) with e=2 or 3 (under a mild assumption on the maximal torus). In characteristic e, the group scheme becomes generically pseudo-reductive and we explain in which sense the F_e[t]-model may still be interpreted as parahoric. Then we focus on the affine Graßmannians (both local and global) attached to this group scheme, which are proved to be representable by an ind-projective ind-scheme. We also obtain normality theorems for Schubert varieties in the local and global case (except if G is an odd dimensional unitary group) and an enumeration of the irreducible components of the fibres via the admissible set. Time permitting, we will explain how in the abelian case these global Schubert varieties give rise to the local models conjectured by Scholze.

16 October 2019 - David Hansen (Max Planck Institute, Bonn) (Note: room change. Seminar is in Huxley room 213.)
Title: Geometric Eisenstein series and the Fargues-Fontaine curve
Abstract: In the geometric Langlands program, one replaces automorphic forms on a group G with sheaves on the stack of G-bundles over a fixed projective curve. The analogue of Eisenstein series in this setting is the "Eisenstein functor" constructed 20 years ago by Braverman-Gaitsgory, which has many marvelous properties. Recently, Fargues has proposed a completely new kind of geometric Langlands program over the Fargues-Fontaine curve. I'll discuss the prospects for constructing an Eisenstein functor in this setting, and explain an application to the local Langlands correspondence. This is joint work in progress with Linus Hamann.

23 October 2019 - Raphaël Beuzart-Plessis (Marseille)
Title: Recent progress on the Gan-Gross-Prasad and Ichino-Ikeda conjectures for unitary groups.
Abstract: In the early 2000s Gan, Gross and Prasad made remarkable conjectures relating the non-vanishing of central values of certain Rankin-Selberg L-functions to the non-vanishing of certain explicit integrals of automorphic forms, called 'automorphic periods' on classical groups. These predictions have been subsequently refined by Ichino-Ikeda and Neal Harris into precise conjectural identities relating these two invariants thus generalizing a famous result of Waldspurger for toric periods for GL(2). In the case of unitary groups, those have now been mostly established by Wei Zhang and others using a relative trace formula approach. In this talk, I will review the story of these conjectures and the current state of the art. Finally, time permitting, I will give some glimpse of the proof.

30 October 2019 - Nadir Matringe (Poitiers)
Title: Galois periods vs Whittaker periods for $SL_n$
Abstract: Let $\pi$ be a generic representation of $SL(n)$, either over a $p$-adic or a finite field, or over the ring of adeles of a number field, in which case we assume $\pi$ to be cuspidal automorphic. In all cases one can characterize representations distinguished by the Galois involution inside the $L$-packet of $\pi$ in terms of nonvanishing of "distinguished" Whittaker periods. We will give an idea of the proofs in each case, and if time allows we will give an application in the adelic setting.

6 November 2019 - James Newton (King’s College)
Title: Symmetric power functoriality for modular forms of level 1
Abstract: Some of the simplest expected cases of Langlands functoriality are the symmetric power liftings Sym${}^r$ from automorphic representations of $GL_2$ to automorphic representations of $GL_{r+1}$. I will discuss some joint work with Jack Thorne on the symmetric power lifting for level 1 modular forms.

13 November 2019 - Jaclyn Lang (Paris 13)
Title: The Hodge and Tate Conjectures for self-products of two K3 surfaces
Abstract: There are 16 K3 surfaces (defined over $\mathbb{Q}$) that Livné-Schütt-Yui have shown are modular, in the sense that the transcendental part of their cohomology is given by an algebraic Hecke character. Using this modularity result, we show that for two of these K3 surfaces $X$, the variety $X^n$ satisfies the Hodge and Tate Conjectures for any positive integer $n$. In the talk, we will discuss the details of the Tate Conjecture for $X^2$. This is joint work in progress with Laure Flapan.

20 November 2019 - Arthur-Cesar Le Bras (Paris 13)
Title : Prismatic Dieudonné theory
Abstract : I would like to explain a classification result for $p$-divisible groups, which unifies many of the existing results in the literature. The main tool is the theory of prisms and prismatic cohomology recently developed by Bhatt and Scholze. This is joint work with Anschütz.

27 November 2019 - Peter Sarnak (Princeton)
Title: Integer points on affine cubic surfaces
Abstract: The level set of a cubic polynomial in four or more variables tend to have many integer solutions, while ones in two variables have a limited number of solutions. Very little is known in case of three variables. For cubics which are character varieties (thus carrying a nonlinear group of morphisms) a Diophantine analysis has been developed and we will describe it. Passing from solutions in integers to integers in, say, a real quadratic field, there is a fundamental change which is closely connected to challenging questions about one-commutators in SL_2 over such rings.

4 December 2019 - Thomas Lanard (Vienna)
Title: On the $l$-blocks of $p$-adic groups
Abstract: We will talk about the category of smooth representations of a p-adic group. Our main focus will be to decompose it into a product of subcategories. When the field of coefficients is $\mathbb{C}$, it is well known thanks to Bernstein decomposition theorem. But when we are over $\bar{\mathbb{Z}}_l$ it is more mysterious. We will see what can be done and some links with the local Langlands correspondence.

11 December 2019 - Elena Mantovan (Caltech)
Title: $p$-adic automorphic forms on unitary Shimura varieties
Abstract: We study $p$-adic automorphic forms on unitary Shimura varieties at any unramified prime $p$. When $p$ is not completely split in the reflex field, the ordinary locus is empty and new phenomena arise. We focus in particular on the construct and study of $p$-adic analogues of Maass--Shimura operators on automorphic forms. These are weight raising differential operators which allow us to $p$-adically interpolate classical forms into families. If time permits, we will also discuss an application to the study of mod $p$ Galois representations associated with automorphic forms. This talk is based on joint work with Ellen Eischen.

The seminar will be preceded by various study groups.

A list of previous seminar talks is here.

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