Jayce Getz (UW-Madion)

Title: A very brief introduction to Shimura varieties and automorphic forms---2 talks

We will give the general definition of a Shimura variety following Deligne, and

then restrict our attention to Shimura varieties "coming from" GL_2. In

particular we will discuss Shimura varieties associated to the unit groups of

quaternion algebras over a totally real field K (this includes the case of

modular curves). Time permitting, we will give the definition of an

automorphic form on GL_2, and perhaps mention Hecke correspondences.

We will attempt to show how modular curves and elliptic modular forms fit into

this framework.

References:

Milne's online course notes (see http://www.jmilne.org/math/):

_Introduction to Shimura Varieties_

_Canonical models of Shimura curves_ (look under "Manuscripts")

For automorphic forms,

Kowalski and Kudla's articles in _Introduction to the Langlands Program_

Birkhauser.

Articles of Piatetski-Shapiro, Borel, Jacquet, et. al. in the "Corvallis notes"

_Automorphic forms, representations, and L-functions_

It's available for free on

http://www.ams.org/online_bks/online_subject.html

_Automorphic Forms and Representations_, Bump

Jeremy Rouse (UW-Madison)

Title: The Weil Conjectures

Reference:

Pierre Deligne. La Conjecture de Weil, I.

Inst. Hautes Etudes Sci. Publ. Math. No. 43

(1974), 273-307.

Chris Holden (UW_Madison)

Title: Mod p representations on elliptic curves" (after Frank

Calegari)

Abstract: Modular Galois representations into GL_2(F_p) with cyclotomic

determinant arise from elliptic curves for p = 2,3,5. We show (by constructing

explicit examples) that such elliptic curves cannot be chosen to have conductor

as small as possible at all primes other than p. Our proof involves finding all

elliptic curves of conductor 85779, a custom computation carried out for us by

Cremona. This leads to a counterexample to a conjecture of Lario and Rio. For p

> 5, we construct irreducible representations with cyclotomic determinant that

do not arise from any elliptic curve over Q.