Ihara's lemma and local Langlands in families

Speaker: 

Claus Sorensen

Institution: 

UC San Diego

Time: 

Tuesday, February 25, 2014 - 2:00pm

Location: 

RH 340P

In their attempt to mimic the proof of Fermat's Last Theorem for GL(n), Clozel, Harris, and Taylor, were led to a conjectural analogue of Ihara's lemma -- which is still open for n>2. In this talk we will revisit their conjecture from a more modern point of view, and reformulate it in terms of local Langlands in families, as currently being developed by Emerton and Helm. At the end, we hope to hint at potential applications.

The rank one abelian Gross-Stark conjecture

Speaker: 

Kevin Ventullo

Institution: 

UC Los Angeles

Time: 

Tuesday, December 3, 2013 - 2:00pm

Location: 

RH 340P

Let $\chi$ be a totally odd character of a totally real number field. In 1981, B. Gross formulated a p-adic analogue of a conjecture of Stark which expresses the leading term at s=0 of the p-adic L-function attached to $\chi\omega$ as a product of a regulator and an algebraic number. Recently, Dasgupta-Darmon-Pollack proved Gross' conjecture in the rank one case under two assumptions: that Leopoldt's conjecture holds for F and p, and a certain technical condition when there is a unique prime above p in F. After giving some background and outlining their proof, I will explain how to remove both conditions, thus giving an unconditional proof of the conjecture. If there is extra time I will explain an application to the Iwasawa Main Conjecture which comes out of the proof, and make a few remarks on the higher rank case.

On the Mumford-Tate conjecture for abelian foufolds

Speaker: 

Bin Zhao

Institution: 

UC Los Angeles

Time: 

Tuesday, November 5, 2013 - 2:00pm to 3:00pm

Location: 

RH 340P

The Mumford-Tate conjecture is a deep conjecture which relates the arithmetic and the geometry of  abelian varieties defined over number fields. The results of Moonen and Zarhin indicate that this conjecture holds for almost all absolutely simple abelian fourfolds. The only exception is when the abelian varieties have no nontrivial endomorphism. In this talk we will begin with an introduction to the Mumford-Tate conjecture and a brief summary of known results towards it. Then we sketch a proof of this conjecture in the above 'missing' case for abelian fourfolds.

Moments of Kloosterman sums and modular forms

Speaker: 

Zhiwei Yun

Institution: 

Stanford University

Time: 

Tuesday, October 29, 2013 - 2:00pm

Location: 

RH 340P

Kloosterman sum is one of the most famous exponential sums
in number theory. It is defined using a prime p (and another number).
How do these sums vary with p? Ron Evans has made several conjectures
relating the moment of Kloosterman sums for p to the p-th Fourier
coefficient of certain modular forms. We sketch a proof of his
conjectures.

A p-adic formula for Chow-Heegner Points

Speaker: 

Michael Daub

Institution: 

University of California at Irvine

Time: 

Tuesday, October 15, 2013 - 2:00pm

Location: 

RH 340P

The Gross-Kudla-Schoen modified diagonal cycle on the triple product of the modular curve with itself provides a wealth of arithmetic information about modular forms, including derivatives of complex L-functions, special values of p-adic L-functions, and points on elliptic curves, known as Chow-Heegner points. In this talk, I will discuss a formula expressing the p-adic logarithm of a Chow-Heegner point in terms of the coefficients of the ordinary projection of certain p-adic modular forms.

Anticyclotomic Iwasawa invariants and congruences of modular forms

Speaker: 

Chan-Ho Kim

Institution: 

University of California at Irvine

Time: 

Tuesday, October 1, 2013 - 2:00pm

Location: 

RH 340P

In this talk, we will look at how congruences between Hecke eigensystems of modular forms affect the Iwasawa invariants of their anticyclotomic p-adic L-functions. It can be regarded as an application of Greenberg-Vatsal's idea on the variation of Iwasawa invariants to the anticyclotomic setting. As an application, we obtain examples of the anticyclotomic main conjecture for modular forms not treated by Skinner-Urban's work. An explicit example will be given.

Heegner points and a B-SD conjecture

Speaker: 

Wei Zhang

Institution: 

Columbia University

Time: 

Tuesday, November 12, 2013 - 2:00pm

Location: 

RH 340P

We prove a B-SD conjecture for elliptic curves (for the p^infinity Selmer groups with arbitrary rank) a la Mazur-Tate and Darmon in anti-cyclotomic setting, for certain primes p. This is done, among other things, by proving a conjecture of Kolyvagin in 1991 on p-indivisibility of (derived) Heegner points over ring class fields. Some applications follow, for example, the p-part of the refined B-SD conjecture in the rank one case.

On L-functions of certain exponential sums

Speaker: 

Jun Zhang

Institution: 

Nankai University, visiting UCI

Time: 

Tuesday, April 23, 2013 - 2:00pm to 3:00pm

Host: 

Daqing Wan

Location: 

RH306

We first briefly review Dwork's trace formula and Wan's decomposition theorems. As an application, we consider a family of Laurent polynomials which is a generalization of the Laurent polynomials appeared in Iwaniec's work, and determine $p$-adic valuations for all the roots of the $L$-functions associated to an Zariski open dense subset of the space of Laurent polynomials. For lower dimension cases, we represent the Zariski open subset explicitly by computing an explicit Hasse polynomial.

Title: p-adic heights and integral points on hyperelliptic curves

Speaker: 

Jennifer Balakrishnan

Speaker Link: 

http://www.math.harvard.edu/~jen/

Institution: 

Harvard University

Time: 

Thursday, May 23, 2013 - 2:00pm

Location: 

RH 306

We give a Chabauty-like method for finding p-adic approximations to
integral points on hyperelliptic curves when the Mordell-Weil rank of
the Jacobian equals the genus. The method uses an interpretation of
the component at p of the p-adic height pairing in terms of iterated
Coleman integrals.  This is joint work with Amnon Besser and Steffen
Mueller.

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