Algebra and Number Theory Seminar Fall 2014

Wednesdays from 2:30-3:30pm

McHenry Library room 4130
For more information please contact Professor Samit Dasgupta or call the Mathematics Department at 831-459-2969

Wednesday, October 8, 2014

"The p-adic Gross-Zagier formula on Shimura curves"

Daniel Disegni, McGill University

For elliptic curves E/Q whose L-function L=L(E,s) vanishes to order one, the rank of E(Q) is also known to be one. This is the first prediction of the Birch and Swinnerton-Dyer conjecture, and the main ingredient of the proof is the formula of Gross and Zagier relating the heights of modularly-constructed (Heegner) points to the central derivative of L. The second prediction of BSD is a formula for the central leading term of L. This is only implied by the Gross-Zagier formula up to a nonzero rational number. One way to go on and study the BSD formula up to p-integrality is provided by a p-adic analogue of the Gross-Zagier formula due to Perrin-Riou and Kobayashi. I will explain this circle of ideas as well as its generalization to totally real fields. Time permitting, I will also discuss the representation-theoretic context.

Wednesday, October 15, 2014

"Locally indecomposable Galois representations with full residual image"

Eknath Ghate, Tata Institute of Fundamental Research

We consider certain p-ordinary non-CM Hida families with full residual Galois image, and give mild conditions under which every classical point of weight at least 2 in these families has a locally indecomposable Galois representation. Our proof uses methods from deformation theory and relies crucially on the existence of certain weight one forms when p = 3. This is joint work with Devika Sharma.

Wednesday, October 22, 2014

"Introduction to Mochizuki's works on inter-universal Teichmuller theory"

Chung Pang Mok, Morningside Center of Mathematics and Purdue University

Inter-universal Teichmuller theory, as developed by Mochizuki in the past decade, is an analogue for number fields of the classical Teichmuller theory, and also of the p-adic Teichmuller theory of Mochizuki. In this theory, the ring structure of a number field is subject to non-ring theoretic deformation. Absolute anabelian geometry, a refinement of anabelian geometry, plays a crucial role in inter-universal Teichmuller theory. In this talk, we will try to give an introduction to these ideas

Wednesday, October 29, 2014

"A Mackey-functor theoretic interpretation of biset functors"

Hiroyuki Nakaoka, Université de Picardie, Amiens

In this talk, I will introduce a 2-category of finite sets with variable finite group actions, which enables us to regard a biset functor as a special kind of Mackey functor on it.
This gives an analog of Dress' definition of a Mackey functor for biset functors.

Wednesday, November 5, 2014

"Mathieu Moonshine and Symmetries of Hyperkaehler manifolds"

Gerald Hoehn, Kansas State University

Mathieu Moonshine started with the observation by Eguchi, Ooguri and Tachikawa in 2010 that the complex elliptic genus of a K3 surface - a weak Jacobi Form of weight 0 and index 1 - encodes simple sums of dimensions of irreducible representations of the Mathieu group M24. I will review the progress which has been achieved so far.
In the second part of my talk, I will relate this observation with the symplectic symmetry groups of K3 surfaces and higher dimensional hyperkaehler manifolds. In particular, I will report on a recent result by G. Mason and myself on the classification of symplectic automorphism groups of hyperkaehler manifolds deformation equivalent to the 2nd Hilbert scheme of a K3 surface.

 Wednesday, November 12, 2014

"The parity conjecture in analytic families"

Jonathan Pottharst

The "parity conjecture" refers to the order of vanishing modulo 2 in the Bloch–Kato conjecture on special values of motivic L-functions.  After specializing to a class of cases where the conjecture can be formulated unconditionally, we generalize techniques of Nekovar to show that the validity of the claim is constant in p-adic analytic families, and give applications to Hilbert modular forms.  This is joint work with Liang Xiao.

 Wednesday, November 19, 2014

"The Eigencurve is Proper"

Hansheng Diao, IAS

The eigencurve is a rigid analytic curve over Q_p parametrizing all finite slope overconvergent modular eigenforms. It is a conjecture of Coleman-Mazur that the eigencurve has "no holes". In other words, the eigencurve is proper over the weight space. We prove that the conjecture is true.

Wednesday, November 26, 2014 TBA

Wednesday, December 3, 2014 TBA

Wednesday, December 10, 2014

"Gal(Q_p-bar/Q_p) as a geometric Galois group"

Jared Weinstein, Boston University

We construct an object defined over an algebraically closed field, whose fundamental group equals the absolute Galois group of Q_p.  Formally, this object is a quotient of a perfectoid space.   We'll motivate its construction by first considering the corresponding object for F_p((t)), which is much simpler.  This object is one facet of a larger program initiated by Peter Scholze.