Overconvergent Eichler-Shimura morphisms for modular curves

Speaker: 

Fucheng Tan

Institution: 

Michigan State University

Time: 

Tuesday, May 20, 2014 - 2:00pm

Location: 

RH 340P

We construct the Eichler-Shimura morphisms for families of overconvergent modular forms via Scholze's theory of pro-etale site, as well as the Hodge-Tate period maps on modular curves of infinite level. We follow some of the main ideas in the work of Andreatta-Iovita-Stevens. In particular, we reprove the main result in their paper. Since we work entirely on the generic fiber of the modular curve, log structures will not be needed if we only consider the Eichler-Shimura morphism for cusp forms.  Moreover, the well-established theory of the Hodge-Tate period map for Shimura varieties of Hodge type may allow us to generalize the construction to more general Shimura varieties. This is a joint work with Hansheng Diao.

Diophantine properties of fields with finitely generated Galois group

Speaker: 

Michael J. Larsen

Institution: 

Indiana University and MSRI

Time: 

Tuesday, April 29, 2014 - 2:00pm

Host: 

Location: 

RH 340P

I will discuss a number of related conjectures concerning the rational points of varieties (especially curves and abelian varieties) over fields with finitely generated Galois group and present some evidence from algebraic numebr theory, Diophantine geometry, and additive combinatorics in support of these conjectures.

Towards a definition of Shimura curves in positive characteristics

Speaker: 

Jie Xia

Institution: 

Columbia University

Time: 

Tuesday, May 13, 2014 - 2:00pm

Location: 

RH 340P

Shimura varieties are defined over complex numbers and generally have number fields as the field of definition. Motivated by an example constructed by Mumford, we find conditions which guarantee a curve in char. p lifts to a Shimura curve of Hodge type. The conditions are intrinsic in positive characteristics and thus they shed light on a definition of Shimura curves in positive characteristics. 

In this talk, I will start with modular curves, and discuss the moduli interpretation of Shimura curves. Then I will present such a condition in terms of isocrystals. Time permitting, I would show a deformation result on Barsotti-Tate groups, which serves as a key step in the proof. 
 

Hecke and Galois Properties of Special Cycles on Unitary Shimura Varieties

Speaker: 

Dimitar Jetchev

Institution: 

EPFL (Lausanne)

Time: 

Tuesday, February 25, 2014 - 4:00pm

Host: 

Location: 

RH 340P

We define a collection of special 1-cycles on certain Shimura 3-folds associated to U(2,1) x U(1,1) and appearing in the context of the Gan--Gross--Prasad conjectures. We study and compare the action of the Hecke algebra and the Galois group on these cycles via distribution relations and congruence relations that would ultimately lead to the construction of a novel Euler system for these Shimura varieties. The comparison is achieved adelically using Bruhat--Tits theory for the corresponding buildings.

Units in function rings

Speaker: 

Daniel Bertrand

Institution: 

Université Pierre et Marie Curie and MSRI

Time: 

Thursday, April 24, 2014 - 2:00pm

Host: 

Location: 

RH 440R

Contrary to their classical namesakes over the ring of integers, Pell equations over function rings in characteristic zero need not have infinitely many solutions. How often this occurs has been the theme of recent work of D. Masser and U. Zannier. The case of smooth curves is governed by the relative Manin-Mumford conjecture on abelian schemes. We pursue this study by considering singular curves and the associated generalized jacobians.

 

p-Adic Artin L-functions

Speaker: 

Ralph Greenberg

Institution: 

University of Washington

Time: 

Tuesday, February 25, 2014 - 3:00pm

Host: 

Location: 

RH 340P

We will discuss the question of defining a p-adic L-function and formulating a main conjecture for an Artin representation. The case where the Artin representation is totally even (or odd) is classical. The corresponding main conjecture has been proven by Wiles.  This talk will discuss the special case where the representation is 2-dimensional, but not totally even or odd. As we will explain, under certain assumptions, there are two p-adic L-functions, two Selmer groups, and two main conjectures. This talk is about joint work with Nike Vatsal. 

The Harder-Narasimhan Filtration on Kisin Modules

Speaker: 

Carl Wang Erickson

Institution: 

Brandeis University

Time: 

Tuesday, April 22, 2014 - 2:00pm

Location: 

RH 340P

We will introduce the notion of slope filtration through examples, including the Harder-Narasimhan filtration on finite flat group schemes due to Fargues. We will then introduce Kisin modules, a certain generalization of finite flat group schemes, and describe a slope filtration on Kisin modules. This is joint work with Brandon Levin. 

The p-adic Eichler-Shimura isomorphism

Speaker: 

Preston Wake

Institution: 

University of Chicago

Time: 

Tuesday, April 8, 2014 - 2:00pm to 3:00pm

Location: 

RH 340P

A theorem of Eichler and Shimura says that the space of cusp forms with complex coefficients appears as a direct summand of the cohomology of the compactified modular curve. Ohta has proven an analog of this theorem for the space of ordinary p-adic cusp forms with integral coefficients. Ohta's result has important applications in the Iwasawa theory of cyclotomic fields.

We discuss a new proof of Ohta's result using the geometry of Hecke correspondences at places of bad reduction.

On mu-invariants and congruences with Eisenstein series

Speaker: 

Rob Pollack

Institution: 

Boston University

Time: 

Tuesday, March 11, 2014 - 2:00pm to 3:00pm

Location: 

RH 340P

For any irregular prime p, one has a Hida family of cuspidal eigenforms of level 1 whose residual Galois representations are all reducible. This family has already played a starring role in Wiles’ proof of Iwasawa’s main conjecture for totally real fields. In this talk, we instead focus on the Iwasawa theory of these modular forms in their own right. We will discuss new phenomena that occur in this residually reducible case including the fact that analytic mu-invariants are unbounded in this family and directly related to the p-adic zeta-function. This is a joint work with Joel Bellaiche.

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