London Number Theory Seminar 



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 padic 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 CalabiYau nfolds converge to an nsphere with respect to the GromovHausdorff metric. Boucksom and Jonsson have shown that, if we choose a family of volume forms on these CalabiYau 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 GromovHausdorff limit. This convergence takes place in a suitable Berkovich space that contains both the complex fibers and the nonarchimedean 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, 12pm in Huxley 642 Oneoff seminar at nonstandard 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 RapoportZink in EL and PEL cases. More recently, a grouptheoretic approach to their definition and study has been possibilitated by the theory of affine Graßmannians, as in the works of PappasRapoport and PappasZhu, 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 quasisplit 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 pseudoreductive 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 indprojective indscheme. 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 FarguesFontaine curve
Abstract: In the geometric Langlands program, one replaces automorphic forms on a group G with sheaves on the stack of Gbundles over a fixed projective curve. The analogue of Eisenstein series in this setting is the "Eisenstein functor" constructed 20 years ago by BravermanGaitsgory, which has many marvelous properties. Recently, Fargues has proposed a completely new kind of geometric Langlands program over the FarguesFontaine 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 BeuzartPlessis (Marseille)
Title: Recent progress on the GanGrossPrasad and IchinoIkeda conjectures for unitary groups.
Abstract: In the early 2000s Gan, Gross and Prasad made remarkable conjectures relating the nonvanishing of central values of certain RankinSelberg Lfunctions to the nonvanishing of certain explicit integrals of automorphic forms, called 'automorphic periods' on classical groups. These predictions have been subsequently refined by IchinoIkeda 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 selfproducts of two K3 surfaces
Abstract: There are 16 K3 surfaces (defined over $\mathbb{Q}$) that LivnéSchüttYui 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  ArthurCesar 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 onecommutators 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
padic 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 MaassShimura 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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