# Number Theory Seminar, Fall 2000

## Thursdays, 10:10-11:00, 115 Osmond

September 7            George Andrews (PSU)

## New Pentagonal NumberTheorems and the Umbral Calculus

September 14           Yuri Zarhin (PSU)

Hyperelliptic jacobians without CM and modular representations

An explicit construction of hyperelliptic jacobians without nontrivial endomorphisms will be presented. Some of our examples use the Steinberg representation of certain finite Chevalley groups.

September 21           Robert Vaughan (PSU)

On sigma-phi numbers

Joint work with Kevin Weis will be described investigating the properties of a sequence of natural numbers similar to, but somewhat easier to study than, Carmichael numbers.

September 28           Irwin Kra (Stonybrook):  MASS Colloquium

Projective embeddings of some modular curves

For fixed values of the parameter $\tau$, theta functions of one variable $\theta[\chi](z,\tau)$ with characteristics $\chi$ can be used to study elliptic function theory. The corresponding theta constants $\theta[\chi](0,\tau)$ can be used to study function theory on surfaces represented by the action of subgroups of the modular group $\Gamma = \mbox{ PSL}(2,{\bf Z})$ on the upper half plane. Theta functions with integer (and, slightly more generally, half integer) characteristics have been studied classically. One is able to get more insight by consideringtheta functions with rational characteristics.

Let $\Gamma(k)$ be the level $k$ (assume for simplicity that $k$ is an odd integer) principal congruence subgroup acting on the upper half plane ${\bf H}^2$. Let $X_k$ be the compactification of ${\bf H}^2/\Gamma(k)$ obtained by filling in the punctures. We discuss projective embeddings of $X_k$ using modifications of theta constants with characteristics of the form $\left [\begin{array}{c} \frac{2l+1}{k} \\ 1 \end{array} \right ]$ and related automorphic forms constructed from derivatives of theta functions.

October 5              Zifeng Yang (PSU)

Zeta Measures over Function Fields

Let A be integral domain of polynomials over a finite field of r elements. A is taken as an analogue of the ring of rational integers, with the "positive" elements in A being the monic polynomials. As an analogue of the zeta function over Z, the zeta function over A can be defined, and it is known that it can be analytically continued to a "complex plane" . In this talk, we will present some recent results in the measure theory of function fields and applications to the zeta function over A.

October 12              Ed Formanek (PSU)

Two questions about free and relatively free groups

Questions of B. Plotkin and A. Myasnikov are answered.
(1)  Let S be the semigroup of endomorphisms of a finitely generated free group.  We show that the only automorphisms of S are the obvious ones.
(2)  Let F(r,c) be a free nilpotent group of rank r and class c.  We determine for which r and c there are nontrivial elements of F(r,c) which are fixed by every automorphism of F(r,c), and for which r and c the automorphism group of F(r,c) has a nontrivial center.

October 19              Winnie Li (PSU)

Coverings of curves with asymptotically many rational points

Infinite families of curves defined over a finite field $F_q$ of $q$ elements containing many rational points can be used to construct good long algebraic geometry codes. Denote by $N_q(g)$ the maximum possible number of rational points contained in a curve of genus $g$ defined over $F_q$. Ihara introduced the quantity $A(q)$ which is the limsup of the  quotient $N_q(g)/g$ as the genus $g$  approaches infinity. Drinfeld and Vladut proved that $A(q) \le \sqrt q$. On the other hand, the work of Ihara, Tsfasman, Vladut and Zink showed that $A(q) = \sqrt q$ when $q$ is a square. Much less is known when $q$ is not a square. In this talk we'll review known optimal constructions and present new lower bounds for $A(q)$ with $q$ nonsquare, in joint work with Hiren Maharaj.

October 24              Thomas Ernst (Uppsala)  (SPECIAL LECTURE:  119 Boucke, 04:00pm)

A new expression for generalized Vandermonde determinants

The purpose of this talk is to present a new expression for the generalized Vandermonde determinant (GVD) ${a}_{\lambda+\delta}$ and thus for the Schur function defined by $s_{\lambda} = \frac{a_{\lambda+\delta}}{a_{\delta}}$.  This expression contains a vector sum of elementary symmetric polynomials.  First the case GVD with two deleted rows is treated and then a formula for an arbitrary GVD is proved by induction.  It would be interesting to try to extend this formula to arbitrary
symmetric polynomials.  In the process we also obtain an equivalence relation on the set of all GVD.

October 26              John Dillon (NSA):  Colloquium

Maps and Character Sums on Finite Fields

Recent research on cyclic difference sets has uncovered some surprising connections with more traditional objects of study such as Gauss and Jacobi sums, Dickson (and exceptional) polynomials and quadratic forms as well as with some fundamental properties of BCH and 1st Order Reed-Muller codes.   We shall discuss a number of these results and raise some questions for further research.

November 2             Andrew Granville (U of Georgia): Colloquium

Distribution  of  values of L(1,chi)

In this talk joint work with Soundararajan will be described one consequence of which is a proof of a conjecture of Montgomery and Vaughan.

November 9             Dale Brownawell (PSU)

Linear independence in positive characteristic and divided derivatives

Function fields in positive characteristic have long provided a rich analogue of number fields.  In particular, Jing Yu has established a full analogue for t-modules of the Baker-Wuestholz Theorem on the linear independence of logarithms in commutative algebraic groups.

Unlike the classical case, the function field case involves a variable in the  base field.  So Laurent Denis had the idea of showing the linear independence of certain derivatives (with respect to this variable) of the logarithm of the most basic t-module, the Carlitz module.  In joint work, we showed the linear independence of all divided derivatives of any non-zero logarithm of an algebraic value for any Drinfeld module.

Recently I was able to show the linear independence of all divided derivatives of all coordinates of any non-zero logarithm of a simple t-module (or a minimal extension thereof).

No background beyond a first-year course in algebra will be assumed.

November 16           Sheeram Abhyankar (Purdue)

Symplectic groups and permutation polynomials

The linear group trinomial provides a mnemonic device for the recently discovered permutation polynomials of M\"uller-Cohen-Matthews, whereas the symplectic group equation generalizes them, thereby giving rise to strong genus zero coverings for characteristic two.

November 30            Prof. Hourong Qin (Nanjing University and Columbia University)

Tame kernels and Tate kernels of quadratic number fields

We give a brief introduction to some connections between algebraic K-theory and number theory. Then we focus on the study of the tame kernels and Tate kernels of quadratic number fields. The results will be applied to the solvability of the Pell's equation.

December 7              Andy Pollington (BYU): Colloquium

Inhomogeneous Diophantine approximation and a conjecture of Barnes and Swinnerton-Dyer concerning indefinite, binary quadratic forms

The Barnes Swinnerton-Dyer states that the inhomogeneous minimum of an indefinite, rational, binary, quadratic form is always rational and isolated in the inhomogeneous spectrum of the form. We confirm this conjecture in certain cases.  The evaluation of such minima is closely related to the class number of real quadratic fields.

December 12            Daniel Bertrand (Paris VI, visiting IAS)
Special lecture, 1:25-2:15, 116 McAllister.

Unipotent representations of classical and differential Galois groups

Unipotent representations of the absolute Galois group of a number field naturally appear in the study of the $l$-adic realizations of a one-motive $M$. In his work on deficient one-motives, K. Ribet showed that the image of these representations become small when $M$ presents an antisymmetric' autoduality.  Similarly, reducible differential operators $L= L_t...L_1$ yield unipotent representations of differential Galois groups. We shall explain how the size of their image reflects the complexity' of the product, and (when $t=3$) that it becomes small iff $L$ presents a 'symmetric' autoduality. We shall further explain how Grothendieck's notion of blended extensions provides a uniform explanation of these phenomena.