## Northwestern Number Theory Seminar Winter 2010

The seminar takes place on Mondays, 3:00-3:50PM in Lunt 107.
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Previous terms: Fall 2008 / Winter 2009 / Spring 2009 / Fall 2009

## Schedule of Talks

Click on title (or scroll down) for the abstract.

 January 4 Le Thai Hoang (UCLA) Arithmetic Combinatorics in Function Fields January 11 (no seminar) January 18 (no seminar: MLK Day) January 25 Frank Calegari (Northwestern) The Galois Groups of Graphs February 1 (no seminar) February 8 (no seminar) February 15 (no seminar) February 22 Kai-Wen Lan (Princeton/IAS) Vanishing theorems for torsion automorphic sheaves March 1 Frank Calegari (Northwestern) Galois representations and the Fontaine-Mazur conjecture March 8 David Geraghty (Harvard) Congruences between weight 2 Hilbert modular forms

## Abstracts

• Le Thai Hoang (UCLA) Arithmetic Combinatorics in Function Fields

Analogies between the integers and the ring F_q[t] have been long known. However, from an arithmetic combinatorics perspective, these analogies have been little and only recently explored. As it turns out, in many cases existing methods can be transfered directly to F_q[t], while at times extra difficulties will arise. In this talk, I will discuss about analogs of some well-known results in this setting, including:

-Green-Tao theorem for F_q[t]: The irreducible polynomials in F_q[t] contain affine spaces of arbitrarily high dimension.

-Sarkozy's theorem for F_q[t]: In any subset of positive density in F_q[t], we can find polynomials f, g such that f-g = h^2 for some nonzero polynomial h in F_q[t].

• Frank Calegari (Northwestern) The Galois Groups of Graphs

Let Gamma be a finite graph. When is the field generated by the largest eigenvalue of the adjacency matrix of G abelian? We study this (and related questions) by investigating some surprising properties of small cyclotomic integers. This is joint work with Noah Snyder and Scott Morrison.

• Kai-Wen Lan (Princeton/IAS) Vanishing theorems for torsion automorphic sheaves

Given a compact PEL-type Shimura variety, a sufficiently regular weight (defined by mild and effective conditions), and a prime number p unramified in the linear data and larger than an effective bound given by the weight, we show that the etale cohomology with Z_p-coefficients of the given weight vanishes away from the middle degree, and hence has no p-torsion. (This is joint work with Junecue Suh.)

• Frank Calegari (Northwestern) Galois representations and the Fontaine-Mazur conjecture

We discuss some general conjectures regarding p-adic Galois representations.

• David Geraghty (Harvard) Congruences between weight 2 Hilbert modular forms

Let F be a totally real field and rhobar an irreducible modular mod l representation of G_F. We prove an existence theorem for potentially Barsotti-Tate modular lifts of rhobar. The key ingredient in the proof is a result guaranteeing the existence of ordinary lifts, after replacing F by a solvable extension. We give applications to modularity lifting theorems for Barsotti-Tate Galois representations. This is joint work with Thomas Barnet-Lamb and Toby Gee.