Forthcoming events in this series


Thu, 27 Sep 2012

11:00 - 12:00
L1

Recovering curves from L-series

Gunther Cornelissen
(Utrecht)
Abstract

The main result of the talk is that two curves over a finite field are isomorphic, up to automorphisms of the ground field, if and only if there is an isomorphism of groups of Dirichlet characters such that the corresponding L-series are all equal. This can be shown by combining Uchida's proof of the anabelian theorem for global function fields with methods from (noncommutative) dynamical systems. I will also discuss how to turn this theorem into a theoretical algorithm that, given a listing of L-functions, determines an equation for the corresponding curve(s).

Thu, 27 Sep 2012

09:30 - 10:30
L1

$\ell$-adic representations of etale fundamental group of curves

Anna Cadoret
(Ecole Polytechnique)
Abstract

I will present an overview of a series of joint works with Akio Tamagawa about l-adic representations of etale fundamental group of curves (to simplify, over finitely generated fields of characteristic 0).
More precisely, when the generic representation is GLP (geometrically Lie perfect) i.e. the Lie algebra of the geometric etale fundamental group is perfect, we show that the associated local $\ell$-adic Galois representations satisfies a strong uniform open image theorem (ouside a `small' exceptional locus). Representations on l-adic cohohomology provide an important example of GLP representations. In that case, one can even provethat the exceptional loci that appear in the statement of our stronguniform open image theorem are independent of $\ell$, which was predicted by motivic conjectures.
Without the GLP assumption, we prove that the  associated local l-adic Galois representations still satisfy remarkable rigidity properties: the codimension of the image at the special fibre in the image at the generic fibre is at most 2 (outside a 'small' exceptional locus) and its Lie algebra is controlled by the first terms of the derived series of the Lie algebra of the image at the generic fibre.
I will state the results precisely, mention a few applications/open questions and draw a general picture of the proof in the GLP case (which,in particular, intertwins via the formalism of Galois categories, arithmetico-geometric properties of curves and $\ell$-adic geometry). If time allows, I will also give a few hints about the $\ell$-independency of the exceptional loci or the non GLP case.

Wed, 26 Sep 2012

11:50 - 12:30
L1

Triple product p-adic L-functions for balanced weights

Matt Greenberg
(University of Calgary)
Abstract

In this talk, I will describe a construction of a $p$-adic L-function attached to a triple of $p$-adic Coleman families of cusp forms.  This function interpolates algebraic parts of special values of Garrett triple product L-functions at balanced triples of weights.  Our construction is complementary to that of Harris and Tilouine which treats the case of unbalanced weights.

Wed, 26 Sep 2012

10:45 - 11:45
L1

An overconvergent  Eichler-Shimura isomorphism

Adrian Iovita
(McGill and Padova)
Abstract

Given a $p$-adic weight and a finite slope we describe a Hecke and Galois equivariant geometric map relating elliptic overconvergent modular symbols and overconvergent modular forms of that slope, appropriate weights and $\mathbf{C}_p$-coefficients. We show that for a fixed slope, with the possible exception of a discrete family of weights, this map is an isomorphism.

Wed, 26 Sep 2012

09:15 - 10:15
L1

The Hodge-Tate sequence and overconvergent $p$-adic modular sheaves

Glenn Stevens
(Boston University)
Abstract

Using Faltings' theory of the Hodge-Tate sequence of an abelian scheme we construct certain sheaves $\Omega^\kappa$, where $\kappa$ is a not-necessarily integral weight, over formal subschemes of modular varieties over which the canonical subgroup exists.   These sheaves generalize the integral powers, $\omega^k$, of the sheaf $\omega$ of relative differentials on a modular curve.   Global sections of $\Omega^\kappa$ provide geometric realizations of overconvergent automorphic forms of non-integral weight.  Applications of this approach to the theory of $p$-adic Hilbert modular forms will be given.   This is joint work with Fabrizio Andreotti and Adrian Iovita.

Tue, 25 Sep 2012

16:45 - 17:25
L1

Radius of convergence of $p$-adic connections and the Berkovich ramification locus

Francesco Baldassarri
(University of Padova)
Abstract

We apply the theory of the radius of convergence of a $p$-adic connection to the special case of the direct image of the constant connection via a finite morphism of compact $p$-adic curves, smooth in the sense of rigid geometry. We show that a trivial lower bound for that radius implies a global form of Robert's $p$-adic Rolle theorem. The proof is based on a widely believed, although unpublished, result of simultaneous semistable reduction for finite morphisms of smooth $p$-adic curves. We also clarify the relation between the notion of radius of convergence used in our previous work and the more intrinsic one used by Kedlaya. (The paper is available athttp://arxiv.org/abs/1209.0081)

Tue, 25 Sep 2012

16:00 - 16:40
L1

On the $p$-adic invariant cycles theorem

Valentina di Proietto
(Tokyo University)
Abstract

For a proper semistable curve over a DVR of mixed characteristics we re prove the ``invariant cycles theorem'' with trivial coefficients  by Chiarellotto i.e. that the group of elements annihilated by the monodromy operator on the first de Rham cohomology group of the generic fiber coincides with the first rigid cohomology group of the special fiber, without the hypothesis that the residue field is finite. This is done using the explicit description of the monodromy operator on the de Rham cohomology of the generic fiber with coefficients convergent F-isocrystals given in a work of Coleman and Iovita. We apply these ideas to the case where the coefficients are unipotent convergent F-isocrystals defined on the special fiber: we show that the invariant cycles theorem does not hold in general in this setting. Moreover we give a sufficient condition for the non exactness. It is a joint work with B. Chiarellotto, R. Coleman and A. Iovita.

Tue, 25 Sep 2012

14:45 - 15:25
L1

A $p$-adic BSD conjecture for modular abelian varieties

Steffen Muller
(University of Hamburg)
Abstract

In 1986 Mazur, Tate and Teitelbaum came up with a $p$-adic analogue of the conjecture of Birch and Swinnerton-Dyer for elliptic curves over the rationals. In this talk I will report on joint work with Jennifer Balakrishnan and William Stein on a generalization of this conjecture to the case of modular abelian varieties and primes $p$ of good ordinary reduction. I will discuss the theoretical background that led us to the formulation of the conjecture, as well as numerical evidence supporting it in the case of modular abelian surfaces and the algorithms that we used to gather this evidence.

Tue, 25 Sep 2012

14:00 - 14:40
L1

Rational torsion points of abelian varieties over a large extension of a local field

Yuichiro Taguchi
(Kyushu University)
Abstract

We extend the following theorem of H. Imai in several ways: If  $A$  is an abelian variety with potentially good reduction over a finite extension  $K$  of  $\mathbf{Q}_p$, then it has only finitely many rational torsion points over the maximal $p$-cyclotomic extension of  $K$. In particular, we prove the finiteness over $K(K^{1/p^\infty})$.

Tue, 25 Sep 2012

11:00 - 12:00
L1

Canonical subgroups via Breuil-Kisin modules

Shin Hattori
(Kyushu University)
Abstract

The overconvergence of the canonical subgroup of the universal abelian variety is one of the key ingredients of the theory of overconvergent modular forms. In this talk, I will show the overconvergence of the canonical subgroup with expected properties via the Breuil--Kisin classification, including the case of $p=2$.

Tue, 25 Sep 2012

09:30 - 10:30
L1

Patching functors and the cohomology of Shimura curves

Toby Gee
(London)
Abstract

I will explain recent joint work with Matthew Emerton and David Savitt, in which we relate the geometry of various tamely potentially Barsotti--Tate deformation rings for two-dimensional Galois representations to the integral structure of the cohomology of Shimura curves. As a consequence, we establish some conjectures of Breuil regarding this integral structure.
The key technique is the Taylor--–Wiles--–Kisin patching argument, which,when combined with a new, geometric perspective on the Breuil–--Mezard conjecture, forges a tight link between the structure of cohomology (a global automorphic invariant) and local deformation rings (local Galois-theoretic invariants).

Thu, 07 Jun 2012

16:00 - 17:00
L3

On the section conjecture in anabelian geometry

Jakob Stix (Heidelberg)
(Heidelberg)
Abstract

The section conjecture of Grothendieck's anabelian geometry speculates about a description of the set of rational

points of a hyperbolic curve over a number field entirely in terms of profinite groups and Galois theory.

In the talk we will discuss local to global aspects of the conjecture, and what can be achieved when sections with

additional group theoretic properties are considered.

Thu, 26 Apr 2012

16:00 - 17:00

Weyl sums for quadratic roots

John Friedlander
(Toronto)
Abstract

We study exponential sums of Weyl type taken over roots of quadratic congruences. We are particularly interested in the situation where the range of summation is small compared to the discriminant of the polynomial. We are then able to give a number of arithmetic applications.

This is work which is joint with W. Duke and H. Iwaniec.

Thu, 01 Mar 2012

16:00 - 17:00
L3

Explicit rational points on elliptic curves

Alan Lauder
(Oxford)
Abstract

I will discuss an efficient algorithm for computing certain special values of p-adic L-functions, giving an application to the explicit construction of

rational points on elliptic curves.

Thu, 16 Feb 2012

16:00 - 17:00
L3

Smooth numbers in arithmetic progressions

Adam Harper
(Cambridge)
Abstract

A number is said to be $y$-smooth if all of its prime factors are

at most $y$. A lot of work has been done to establish the (equi)distribution

of smooth numbers in arithmetic progressions, on various ranges of $x$,$y$

and $q$ (the common difference of the progression). In this talk I will

explain some recent results on this problem. One ingredient is the use of a

majorant principle for trigonometric sums to carefully analyse a certain

contour integral.

Thu, 19 Jan 2012

16:00 - 17:00
L3

New perspectives on the Breuil-Mézard conjecture

Toby Gee
Abstract

I will discuss joint work with Matthew Emerton on geometric
approaches to the Breuil-Mézard conjecture, generalising a geometric
approach of Breuil and Mézard. I will discuss a proof of the geometric
version of the original conjecture, as well as work in progress on a
geometric version of the conjecture which does not make use of a fixed
residual representation.