## EVENT DETAILS AND ABSTRACT

**Number Theory**
**Title:** Ramification in crystalline p-adic Galois representations

**Speaker:** Bryden Cais

**Speaker Info:** Univ. of Arizona, Tuscon

**Brief Description:**

**Special Note**:

**Abstract:**

For a prime p and an abelian scheme A over the ring of integers O_K in a number field K, the p-adic Tate module V_pA provides a continuous linear representation \rho_A of the absolute Galois group G_K with coefficients in the field Q_p of p-adic numbers. This representation encodes a lot of the geometry and arithmetic of A, and is of serious interest in number theory. As G_K is (topologically) generated by decomposition subgroups {D_q} for q ranging over the primes of O_K, it is natural to study the restriction of \rho_A to each D_q. If q does not divide p, then this restriction is unramified, so is determined by a single matrix (the value on a Frobenius element). However, when q lies over p, the restriction of \rho_A to D_q is always deeply ramified (the image of the wild inertia subgroup is infinite), and it is natural to try and understand the nature of this wild ramification. In this talk, we will generalize a theorem of Kisin (independently proved by Beilinson and first conjectured by Breuil) which says, in a way that will be made precise, that crystalline p-adic representations (of which the restriction of \rho_A to D_q for q lying over p is an example) nonetheless behave a lot like unramified representations. This is joint work with Tong Liu.

**Date:** Monday, November 17, 2014

**Time:** 4:00PM

**Where:** Lunt 107

**Contact Person:** Patrick Allen

**Contact email:** pballen@math.northwestern.edu

**Contact Phone:**

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