- September 4, 2003,
**Ahmad El-Guindy (UW-Madison)**

Title:*Weierstrass Points on X_0(pM) and Supersingular j-invariants"*

Abstract: Weierstrass points are special points on a Riemann surface that carry a lot of information. Ogg studied such points on X_0(pM) (for M such that the genus of X_0(M)is 0 and prime p not dividing M) and proved that the reduction of Weierstrass points on X_0(pM) is supersingular mod p. In this talk we show that, for square free M on the list, all supersingular j-invariants are covered this way. Furthermore, In most cases where M is prime we describe the explicit correspondence between Weierstrass points and supersingular j-invariants. Along the way we also genarlize a useful formula of Rohrlich for computing a certain Wronskian of modular forms modulo p. - September 11,
**Nigel Boston (UW-Madison)**

Title:*Bounds on cyclic codes using arithmetical geometry*

Abstract: There are many results on the minimum distance of a cyclic code of the form that if a certain set T is a subset of the defining set of the code, then the minimum distance of the code is greater than some integer t. This includes the BCH, Hartmann-Tzeng, Roos, and shift bounds and generalizations of these. In this talk, we define certain projective varieties V(T,t) whose properties determine wh ether, if T is in the defining set, the code has minimum distance exceeding t. Thus our attention shifts to the study of these varie ties. By investigating them using class field theory and arithmetical geometry, we will prove various new bounds. It is interesting, however, to note that there are cases that existing methods handle, that our methods do not, and vice versa. We end with a number o f conjectures. No prior familiarity with codes is assumed. - September 18, Tonghai Yang (UW-Madison)

Title:*The Gross-Keating invariants of a quadratic form*

Abstract: For a quadratic form over Q_p with p \ne 2, its Jordan form is unique in some sense. This is not true when p=2. Gross and Keating define some mysterious invariants in this case, which is magically unifying certain formulas related to quadratic forms, such as local densities. In this talk, we will introduce these invariants, how to compute them in some cases, and motivation for study these invariants. The question is wide open in understanding these invariants. - September 25,
**Eric Bach (UW-Madison)** - October 2,
**Ken Ono (UW-Madison)**

Title:*Parity of the partition function* - October 9,
**Brian Conrad (Univ. Michigan at Ann Arbor)**

Title:*Modular curves and Ramanujan's continued fraction*

Abstract: In Ramanujan's 1916 letter to Hardy, he proposed some remarkable identities that were a special case of a general q-series continued fraction that he had discovered and investigated. Ramanujuan's continued fraction has been the subject of intense scrutiny by many people who have investigated its many identities and the algebraic properties of special values. However, none of this work addresses the most important question of all: is there an underlying principle that explains why Ramanujan's continued fraction has such properties? We provide an affirmative answer to this question by proving that Ramanujan's function (or rather, its reciprocal) is related to the modular curve X(5) as j is to X(1). Moreover, we prove that this link with X(5) is extremely well-behaved with respect to two arithmetic models of X(5), and by means of such a connection with algebraic geometry we are able to give clean conceptual proofs of results that require a lot of algebraic effort from the q-series point of view. For example, we give a new proof that the values of Ramanujan's function at imaginary quadratic points are algebraic integral units, we propose a new algorithm for rapid computation of special values, and we use positive-characteristic geometry to develop an analogue of Kronecker's congruence formula (going far beyond the few results obtained on a case-by-case basis via the q-series methods). This is joint work with Bryden Cais. - October 16,
**David Manderscheid (U. Iowa)**

Title:*Waldspurger's Involution and Types.*

Abstract: In this talk I will parametrize Waldspurger's involution for supercuspidal representations in terms of types in the case p odd. This involution is defined on the discrete series representations of the non-trivial two-fold cover of SL(2,F) where F is a p-adic field. Waldspurger used this involution in his deep study of automorphic forms on the two-fold cover of SL(2) over a number field. In general, types give inducing data that contains the sort of arithmetic information necessary to study the fundamental role that supercuspidal representations play in questions in representation theory of number-theoretic interest, especially functoriality. In the case at hand, types provide a rather surprising result in contrast to the case of SL(2). I will keep prerequisites to a minimum, provide motivation, and try to make this talk understandable for graduate students. - October 23 Two speakers this week.

**Yangbo Ye (U. Iowa) at 11am**

Title:*Selberg's orthogonality for automorphic L-functions*

Abstract: Selberg's orthogonality conjecture predicts that the coefficients of automorphic L-functions attached to different cuspidal representations are orthogonal. Professor Jianya Liu and I first proved a weaker, weighted version of this conjecture. As an application, we then proved that if an L-function can be factored into a product of L-functions of possibly different GL(m) over Q, then this factorization is unique. In other words, we proved the uniqueness of functoriality in this case. In particular, an L-function attached to a cuspidal representation of GL(m) over Q cannot be factored further. The proofs are unconditional. Next we proved the original version of Selberg's orthogonality conjecture under the generalized Ramanujan conjecture. Our results can be used to characterize asymptotically whether two cuspidal representations are equivalent, twisted equivalent, or not twisted equivalent at all. This proof also allowed us to study statistical correlations between non trivial zeros of two automorphic L-functions, under the Ramanujan conjecture. Our proofs are unconditional when the representations are corresponding to holomorphic cusp forms for GL(2).

**Jianya Liu (Shandong Univ., China) at 2:25pm**

Title:*SUBCONVEXITY FOR RANKIN-SELBERG L-FUNCTIONS OF MAASS FORMS*

Abstract: This is a joint work with Yangbo Ye. We prove a subconvexity bound for Rankin-Selberg L-functions L(s,f x g) associated with a Maass cusp form f and a fixed cusp form g in the aspect of the Laplace eigenvalue 1/4+k^2 of f, on the critical line Re s=1/2. Using this subconvexity bound, we prove the equidistribution conjecture of Rudnick and Sarnak on quantum unique ergodicity for dihedral Maass forms. Also proved here is that the generalized Lindelof hypothesis for the central value of our L-function is true on average. - October 30
**Fernando Rodriquez Villegas (U. Texas at Austin)**

Title:*On the E-polynomial of certain character variety*

Abstract: This is work in progress joint with Tamas Hausel (UT Austin). The character variety of the title is a twisted form X of the variety parameterizing the representations of the fundamental group of a Riemann surface of genus g into GL_n. We expect to obtain a description of its E-polynomial (a cohomological invariant of X) by counting the number of points of X over finite fields. Eventually, we would like to use this technique to prove a mirror symmetry type conjecture of Hausel and Thaddeus relating the corresponding character varieties with, respectively, PGL_n and SL_n in place of GL_n. In this talk I will describe the calculation of a generating function giving the zeta functions of X for all n over a given finite field. - November 6
**Jeremy Teitelbaum (UIC)**

Title:*p-adic Fourier Theory and Lubin-Tate Groups*

Abstract: In this talk I will discuss some results (joint with Peter Schneider) on p-adic integration, generalizing old results of Amice and Lazard. Let $L$ be a finite extension of ${\bf Q}_p$. A locally $L$-analytic function on the ring of integers $o=o_L$ is a function given locally by a convergent power series in one variable. We study the space of these functions and its dual, the ring $D(o_L)$ of locally $L$-analytic distributions. We show that this ring of distributions is isomorphic to the global functions on a rigid space $\hat{o}$ parameterizing $L$-analytic characters of $o$. The space $\hat{o}$ turns out to be quite interesting. If $L=\Qp$, then Amice and Lazard showed that $\hat{o}$ is the open unit disk (viewed as a rigid space). We show that, if $L$ is not ${\bf Q}_{p}$, then $\hat{o}$ is a "twisted form" of the open unit disk; it is isomorphic over ${\bf C}_{p}$ to the open unit disk, but is not a disk over any discretely valued extension of $L$. Our methods rely on Lubin-Tate theory and some results from Tate's classic paper on p-divisible groups. We will mention some applications to representation theory and to p-adic L functions. - November 13
**Alex Popa (Princeton University)**

Title:*Central values of Rankin L-series over real quadratic fields*

Abstract: We consider the L-function of a classical modular form f of weight 2 or 0 over the rational numbers, twisted by a Hecke character of a real quadratic field. When the sign of the functional equation is +1, we give an explicit formula for the central value of the L-function in terms of a "toric integral" of a modular form on a quaternion algebra related to f by the Jacquet-Langlands correspondence. The proof uses techniques from the theory of automorphic representations, in particular an adelic version of the Rankin-Selberg integral, and an explicit computation of the theta correspondence between representations of GL(2) and representations of certain similitude groups associated to 2 and 4 dimensional quadratic spaces. - November 20, 2003,
**Jordan Ellenberg (Princeton Univ.)**

Title:*Arithmetic of towers of algebraic curves*

Abstract: "Let ...-> X_n -> ... -> X_0 = X be a tower of curves over a fixed number field F, whose Galois group is a p-adic Lie group (for instance, the tower of Fermat curves x^{p^n} + y^{p^n} + z^{p^n} = 0, or the tower of modular curves X(p^n)). One is interested in general in the variation of arithmetic invariants of curves in such a tower; in particular, we study the variation of the Mordell-Weil rank of the Jacobian of X_n as n grows. We show that this rank grows at most linearly in genus(X_n), and describe how to bound the constants in special cases. The problem has an evident Iwasawa-theoretic flavor, and indeed we show how to package the Mordell-Weil groups of the X_n into an Iwasawa module for a certain non-abelian Iwasawa algebra, of the type recently studied by Coates, Schneider, Sujatha, Howson, Venjakob, and others." - November 27 (Thanksgiving, no seminar)
- December 4
**Paul Jenkins (UW-Madison)**

Title:*Traces of Singular Moduli*

Abstract: We use the circle method to examine the difference between the trace T(d) of a singular modulus coming from the ideal class group of Q(sqrt(-d)) and the estimate given by the first term of the j-function and the elements of the ideal class group.

# Spring 2004 Number Theory Seminar Schedule

Thursdays at 1:20 pm, - Jan. 22,
**Romyar Sharifi (Max-Planck Inst. for Math., Bonn)** - Jan. 29,
**Ralf Schmidt (U. Minnesota)**

Title:*"Local newforms for GSp(4)"*

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,
- Feb. 12,
- Feb. 19,
- Feb. 26,
**Stephen Wainger (UW-Madison)** - Mar. 4,
**Stephen Wainger (UW-Madison)** - Mar. 11,
**Rob Benedetto (Amherst College)** **Mar. 18,****Spring Break****Mar. 25,****Mar. 31****April 8, 2004,****Wenzhi Luo (The Ohio State University)****Apr. 15, 2004,****Ling Long (Iowa State U.)****Apr. 22,****George Pappas (Michigan State U.)****Apr. 29,****Mark Kisin (U. Chicago)****May 6,**