# Spring 2004 Number Theory Seminar Schedule

Thursdays at 1:20 pm, B309 Van Vleck.
• Jan. 22, Romyar Sharifi (Max-Planck Inst. for Math., Bonn)

• Title: Iwasawa theory for Kummer extensions
Abstract: Classical Iwasawa theory takes place over Z_p-extensions of a number field F, such as the cyclotomic Z_p-extension K. The main result of classical Iwasawa theory, due to Mazur and Wiles, is Iwasawa's Main Conjecture, which describes the structure of the maximal unramified abelian pro-p extension of K in the case of F abelian. We will discuss Iwasawa theory for extensions L/F with Galois group isomorphic to a semi-direct product of two copies of Z_p, the quotient Z_p-extension being the cyclotomic Z_p-extension K. I will summarize three very different results on the structure of the maximal unramified abelian pro-p extension X of L. The first illustrates that X can be large when L is a CM-field (with Hachimori), the second relates the structure of X to Massey products, and the third compares X for a particular L to the Eisenstein ideal in Hida's ordinary Hecke algebra.
• Jan. 28, Chian-Jen Wang (U. Minnesota) ( Special time and place)

• Title: Distinguished representations of the metaplectic group
Time and Place: Wed. Jan. 28 at 2:25pm, VV B105
Abstract: An automorphic representation of the metaplectic cover of GL is called "distinguished" if it has a unique Whittaker model. Distinguished representations can be viewed as generalizations of classical theta functions. In 1984, Patterson and Piatetski-Shapiro constructed cuspidal distinguished representations on the three-fold covers of GL(3) using the method of the converse theorem. In this talk, I will discuss recent progress toward generalizing the work of Patterson and Piatetski-Shapiro to the case of the four-fold cover of GL(4).
• Jan. 29, Ralf Schmidt (U. Minnesota)

• Title: "Local newforms for GSp(4)"
Abstract: I will report on a joint project with Brooks Roberts concerning local newforms for the p-adic group $GSp(4)$. Let $\pi$ be an irreducible, admissible generic representation of $PGSp(4,F)$, where $F$ is a p-adic field. We conjecture that (i) There exists an N such that $\pi$ contains a vector invariant under the paramodular group of level N. (ii) If N is minimal with this property, then such a vector is unique up to multiples; we call it a local newform. (iii) The Novodvorski zeta integral of the local newform computes the L-factor of $\pi$. In other words, we conjecture that there is a newform theory analogous to the well-known theory for $GL(2)$. There is another conjecture concerning the structure of oldforms. I will report on recent progress concerning these conjectures.
• Feb. 5, Matthew Boylan (UIUC)

• Title: "Coefficients of half-integral weight modular forms modulo l^j."
Abstract: Suppose that l>=5 is prime, that j>=0 is an integer, and that F(z) is a half-integral weight modular form with integral Fourier coefficients. We give some general conditions under which the coefficients of F are well-distributed'' modulo l^j. As a consequence, we settle many cases of a classical conjecture of Newman by proving, for each prime power l^j with l>= 5, that the ordinary partition function p(n) takes each value modulo l^j infinitely often. This is joint work with Scott Ahlgren.
• Feb. 12, Michael Bush (UIUC)

• Title: Maximal unramified $p$-extensions and $p$-groups of deficiency zero
Abstract: Let $p$ be an odd prime and $k^{nr,p}$ the maximal unramified $p$-extension of an imaginary quadratic field $k$. After reviewing some results of Koch and Venkov we will show that the extension $k^{nr,p}/k$ is never metacyclic. We will then describe some recent computations when $p = 3$ and write down pro-$3$ presentations for two families of groups, one of which may have some bearing on an open question regarding $p$-groups of deficiency zero.
• Feb. 19,
• Feb. 26, Stephen Wainger (UW-Madison)

• Title: Application of the circle method to some problems in Harmonic Analysis and Ergodic Theory
• Mar. 4, Stephen Wainger (UW-Madison)

• Title: Application of the circle method to some problems in Harmonic Analysis and Ergodic Theory II
• Mar. 11, Rob Benedetto (Amherst College)

• Title: Recent Developments in p-adic Dynamics
Abstract: Let K/Q_p be a finite extension of the p-adics, and let f(z) be a rational function in K(z). We will consider iterates f^n (that is, f composed with itself n times). The periodic and preperiodic points of f are analogous to the torsion points on an elliptic curve; to understand them, we will study the dynamics of iterates of f on the p-adic sphere P^1(C_p). In particular, we will present some recent results on recurrent critical points and on wandering domains.
• Mar. 18, Spring Break
• Mar. 25, Akshay Venkatesh (MIT)

• Title: 3-torsion in class groups and counting points on curves
Abstract: I shall discuss some nontrivial bounds on the size of the 3-torsion part of class groups of quadratic fields, and give some applications. This answers a question raised by Brumer-Silverman. This is derived from a method of counting points on curves which is of independent interest. Joint work with H. Helfgott.
• Mar. 31, Tonghai Yang (UW-Madison)

• Title: Number fields and modular forms
• April 8, 2004, Wenzhi Luo (The Ohio State University)

• Title: A refined non-vanishing theorem for automorphic L-functions on GL(n)
Abstract: Given any point on the complex plane, and any unitary cuspidal automorphic representation of GL(2) over the rationals, there exist some twists (by Dirichlet characters) of the associated standard L-function which do not vanish at this point, by a theorem of D.Rohrlich. We generalize this to GL(3). Moreover for n > 3, our method yields better non-vanishing results than the currently known bounds obtained by L.Barthel and D. Ramakrishnan.
• Apr. 15, 2004, Ling Long (Iowa State U.)

• Title: "On cusp forms of noncongruence subgroups"
Abstract: Theory of modular form of noncongruence subgroups is far less developed compared to theory of congruence modular form. The Atkin-Swinnerton-Dyer conjecture is one of the major conjectures for cusp forms of noncongruence subgroups. While there has been significant progress, for example, the weak Atkin-Swinnerton-Dyer conjecture proved by Scholl, the full version of the conjecture still remain to be open. We are going to discuss a particular situation where some spaces of cusp forms of nongongruence subgroups admit bases satisfying the Atkin-Swinnerton-Dyer congruence relations with certain newforms of congruence subgroups. Consequently, the Atkin-Swinnerton-Dyer conjecture for these spaces are established.
• Apr. 22, George Pappas (Michigan State U.)
Title: The Steinitz class of weight 2 cusp forms"
Abstract: We will explain how to calculate  the Steinitz class of the lattice of cusp forms of weight 2 with integral Fourier coefficients and fixed Nebentypus character.
He will also give a colloquium on Friday, April 23 at 4pm. Here is the title and abstract
Title: Geometric Galois modules"
Abstract: Consider a Galois cover X/Y of algebraic varieties defined over the integers with finite Galois group G. We will discuss work towards determining various cohomology groups of X as G-modules and explain interesting connections with L-functions and the theory of ideal class groups of cyclotomic fields.
• Apr. 29, Mark Kisin (U. Chicago)

• Title: Locally analytic cuspidal representations for GL_2(Q_p)
Abstract: In the usual theory of smooth representations of an algebraic group G over Q_p one studies irreducible representations which arise in the space of locally constant functions on G. One can instead consider representations which arise in the space of locally analytic functions on G. I will explain joint work with Matthias Strauch where we construct families of such representations for GL_2, which in a certain sense, p-adically interpolate smooth supercuspidal representations of GL_2. This builds on previous work of Morita and Schneider-Teitelbaum who did the analogous thing for the principal series. These locally analytic representations are expected to be the p-adic local factors in certain global objects which are obtained by p-adically interpolating modular forms.
• May 6, Chia-Fu Yu (Columbia University)

• Title: "Fine structures of Hilbert-Blumenthal varieties and Hecke orbit problems".
Abstract: "We will present the recent progress on the geometry of the reduction of Hilbert-Blumenthal varieties. Especially, we will talk about the dimension, singularieties, and the irreducibility of some $p$-adic invariant strata. We will also present the current results on the Hecke orbit problems on Hilbert-Blumenthal varieties. Most of this talk is joint work with Ching-Li Chai."