Number Theory Seminar References---Tuesdays, Fall 05


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.