Tag - Analytic number theory

Xuhua He: Basic loci of Shimura varieties

6th April, 2017

In mod-p reductions of modular curves, there is a finite set of supersingular points and its open complement corresponding to ordinary elliptic curves. In the study of mod-p reductions of more general Shimura varieties, there is a "Newton stratification" decomposing the reduction into finitely many locally closed subsets, of which exactly one is closed. This closed set is called the basic locus; it recovers the supersingular locus in the classical case of modular curves. In certain cases, the basic locus admits a simple description as a union of classical Deligne-Lusztig varieties. The precise description in these case has proved to be useful for several purposes: to compute intersection numbers of special cycles and to prove the Tate conjecture for certain Shimura varieties. We will describe a group-theoretic approach to understand this phenomenon. We will show that this phenomenon is closely related to the Hodge-Newton decomposition, and many other nice properties on the Shimura varieties. This talk is based on the joint work with Ulrich Gortz and Sian Nie.

Arno Kret: Galois Representations for the general symplectic group

30th March, 2017

In a recent preprint with Sug Woo Shin I construct Galois representations corresponding for cohomological cuspidal automorphic representations of general symplectic groups over totally real number fields under the local hypothesis that there is a Steinberg component. In this talk I will explain some parts of this construction that involve the eigenvariety.

Yihang Zhu: The Hasse-Weil zeta functions of the intersection cohomology of minimally compactified orthogonal Shimura varieties

20th October, 2016

Initiated by Langlands, the problem of computing the Hasse-Weil zeta functions of Shimura varieties in terms of automorphic L-functions has received continual study. We will discuss how recent progress in various aspects of the field has allowed the extension of the project to some Shimura varieties not treated before. In the particular case of orthogonal Shimura varieties, we discuss the computation of the Frobenius-Hecke traces on the intersection cohomology of their minimal compactifications, and the comparison to the Arthur-Selberg trace formula via the process of stabilization. Key ingredients include comparing Harish Chandra character formulas to Kostant's theorem on Lie algebra cohomology, and a comparison between different normalizations of the transfer factors for real endoscopy to get all the signs right.

Douglas Ulmer: Rational curves on elliptic surfaces

5th May, 2016

Given a non-isotrivial elliptic curve E over K=𝔽q(t), there is always a finite extension L of K which is itself a rational function field such that E(L) has large rank. The situation is completely different over complex function fields: For "most" E over K=ℂ(t), the rank E(L) is zero for any rational function field L=ℂ(u). The yoga that suggests this theorem leads to other remarkable statements about rational curves on surfaces generalizing a conjecture of Lang.

Junecue Suh: Standard conjecture of Künneth type with torsion coefficients

21st April, 2016

p>A.Venkatesh asked us the question, in the context of torsion automorphic forms: Does the Standard Conjecture (of Grothendieck's) of Künneth type hold with mod p coefficients? We first review the geometric and number-theoretic contexts in which this question becomes interesting, and provide answers: No in general (even for Shimura varieties) but yes in special cases.

Michael Harris: Modularity and potential modularity theorems in the function field setting

17th December, 2015

Let G be a reductive group over a global field of positive characteristic. In a major breakthrough, Vincent Lafforgue has recently shown how to assign a Langlands parameter to a cuspidal automorphic representation of G. The parameter is a homomorphism of the global Galois group into the Langlands L-group LG of G. I will report on my joint work in progress with Böckle, Khare, and Thorne on the Taylor-Wiles-Kisin method in the setting of Lafforgue's correspondence. New (representation-theoretic and Galois-theoretic) issues arise when we seek to extend the earlier work of Böckle and Khare on the case of GLn to general reductive groups. I describe hypotheses on the Langlands parameter that allow us to apply modularity arguments unconditionally, and I will state a potential modularity theorem for a general split adjoint group.

Jeremy Booher: Geometric Deformations of Orthogonal and Symplectic Galois Representations

19th November, 2015

For a representation of the absolute Galois group of the rationals over a finite field of characteristic p, we would like to know if there exists a lift to characteristic 0 with nice properties. In particular, we would like it to be geometric in the sense of the Fontaine-Mazur conjecture: ramified at finitely many primes and potentially semistable at p. For 2-dimensional representations, Ramakrishna proved that under technical assumptions, odd representations admit geometric lifts. We generalize this to higher dimensional orthogonal and symplectic representations. The key ingredient is a smooth local deformation condition obtained by analysing unipotent orbits and their centralizers in the relative situation, not just over fields.

Patrick Allen: Adjoint Selmer groups for polarized automorphic Galois representations

15th October, 2015

Given the p-adic Galois representation associated to a regular algebraic polarized cuspidal automorphic representation, one naturally obtains a pure weight zero representation called its adjoint representation. Because it has weight zero, a conjecture of Bloch and Kato says that the only de Rham extension of the trivial representation by this adjoint representation is the split extension. We will discuss a proof of this case of their conjecture, under an assumption on the residual representation. This is done by using the Taylor-Wiles patching method, Kisin's technique of analyzing the generic fibre of deformation rings, and a characterization of smooth closed points in the generic fibre of certain local deformation rings.