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

References for first talk:

1. A. Weil, Number of solutions of equations in finite fields, Bull. Am.
Math. Soc., 55 (1949), 497-508.

2. P. Deligne, Formes modulaires et representation l-adiques. Seminaire
Bourbaki, 355 : 139-172.

3. B. Conrad, Modular Forms, Cohomology, and the Ramanujan Conejecture.
Unpublished. E-mail me if you want a copy.

References for second talk:

1. J. Milne, Lectures on Etale Cohomology, course notes. Available online
at http://www.jmilne.org/math/.

2. R. Hartshonre, Algebraic Geometry, Graduate Texts in Mathematics, No.
52, Springer-Verlag, 1977. Sections 2.1, 3.1, and Appendix C.

3. P. 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.

Frank Thorne (UW-Madison)

Title: Ramanujan's Ternary Quadratic Form

Abstract: We discuss a theorem of Ono and Soundararajan, that conditional on the Generalized Riemann Hypothesis and the Conjecture of Birch and Swinnerton-Dyer, every odd integer above 2,719 is represented by the quadratic form x^2 + y^2 + 10 z^2. We begin by giving an overview of the general theory of quadratic forms, and then discuss in detail Ono and Soundararajan's proof for the special case of N squarefree (which does not depend on GRH or BSD). We conclude by giving an overview of the remainder of Ono and Soundararajan's proof.

Ben Kane (UW-Madison)

Title: Bounding Special Values, Twists, and Tate-Shafarevich.

Abstract: In this talk, we will investigate the importance of bounding special
values of L-functions and also bounding twists of these L-functions. In
particular, we will see how Goldfeld, Lieman, and Szpiro used
these techniques to bound the size of the Tate-Shafarevich group, assuming
certain conjectures. Due to some wonderful theorems, some of these
results even hold unconditionally in certain cases.

Matt Darnall (UW-Madison)

Title: The Picard group

In this talk I will describe the various defintions of the Picard Group and show in what cases these defintions are equivalent. I will also try to give some background on some of the words Jordan uses in his class, but never completely defines.

Rob Rhoades (UW-Madison)

Title: p-adic and Combinatorial Properties of Modular Form Coefficients.

Abstract: A well known result is that if E_{2k} is the Eisenstein series of weight 2k and 2k = 2k' modulo (p-1)p^r, then E_{2k} =E_{2k'} modulo p^{r+1}. In words, this result tells us that the Eisenstein series form a natural family of modular forms that are p-adically interpolated in the weight aspect.? Motivated by a question of Serre, we construct a second natural family of modular forms that have this same property.? Before proving such a result we will build up the necessary machinery by demonstrate congruences between the coefficients of two infinite families of modular forms and combinatorially defined objects.

This is joint work with Po-Ru Loh of Caltech.

Patrick Rault (UW-Madison)

Title: Arakelov Theory

Abstract: Arakelov Theory studies the analogy between Number Fields (finite extensions of Q) and Function Fields of Curves over Finite Fields (finite extensions of F_q(t)). In both cases we have a notion of divisor, divisor class group Pic^0, and a Riemann-Roch theorem. In this talk we will discover the theory's most quoted result: a proof, due to Iwasawa and Tate, of Dirichlet's Unit Theorem and the finiteness of the ideal class group of a number field, Cl_F, without the use of Minkowski Theory.

David Dueber (UW-Madison)

Title: Lower Bounds on Discriminants

Abstract: Prior to 1974-ish, all lower bounds for discriminants of number fields based on their degree came from the geometry of numbers, and were mostly only refinements of work by Minkowski. Then Stark proved an "effective" version of the Brauer-Siegel Theorem with which Odlyzko applied to create
several formulas for the discriminant, from which lower bounds can be obtained. I will talk both about proving this effective Brauer-Siegel Theorem (or possibly a somewhat related result of Heilbronn's), and also work through some of the techniques involved in the estimation of Odlyzko's formulas.