# 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**

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**"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.

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**Wednesday, October 15, 2014**

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**"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**

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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

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*Introduction to Mochizuki's works on inter-universal Teichmuller theory*"Chung Pang Mok, Morningside Center of Mathematics and Purdue University

**Wednesday, October 29, 2014 **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.

*"A Mackey-functor theoretic interpretation of biset functors"*

**Hiroyuki Nakaoka, Université de Picardie, Amiens**

This gives an analog of Dress' definition of a Mackey functor for biset functors.

**Wednesday, November 5, 2014 **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.

*Gerald Hoehn, Kansas State University*

**"Mathieu Moonshine and Symmetries of Hyperkaehler manifolds"**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" 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.

*"The parity conjecture in analytic families"*Jonathan Pottharst

** Wednesday, November 19, 2014 **

*Hansheng Diao, IAS*

**"The Eigencurve is Proper"**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.