Number Theory Seminar, Spring 2001

Thursdays, 11:15-12:05, 103 McAllister

January 11         Matt Papanikolas (Brown University)

Periods of Drinfeld modules with complex multiplication

We investigate transcendence properties of periods of Drinfeld modules with complex multiplication.  In particular we show that if such Drinfeld modules have different CM fields then their fundamental periods are algebraically independent over the algebraic numbers.  Joint work with Dale Brownawell.

January 25          Robert Vaughan (PSU)

Waring's problem: A Survey

Recent work on G(k) in Waring's problem, jointly with T. D. Wooley, will be described and placed in a historical context.  Some speculations will be made about future directions in the Hardy-Littlewood method.

February 1          No seminar:  Peter Sarnak (Princeton University and the Institute for Advanced study) is giving the Russell Marker Lectures in Mathematics

L-Functions, Arithmetic and Semiclassics

Monday, January 29, 8:00 p.m., 112 Osmond Laboratory, Hilbert's Eleventh Problem

Tuesday, January 30, 4:30 p.m., 112 Osmond Laboratory, Lp Norms of Eigenfunctions

Wednesday, January 31, 4:30 p.m., 112 Osmond Laboratory, Quantum Unique Ergodicity

Thursday, February 1, 4:30 p.m., 112 Osmond Laboratory, Families of L-Functions and Symmetry

February 8          Sanju Velani (Queen Mary Westfield College, London), 103 McAllister

On simultaneously badly approximable pairs

For any pair $i,j \geq 0$ with $i+j =1$ let $\Bad(i,j)$ denote the set of  pairs $(\a,\b) \in \R^2$ for which $ \max \{ ||q\a||^{1/i}\, ||q\b||^{1/j} \} > c/q $ for all $ q \in \N $. Here $c = c(\a,\b)$ is a positive constant. If $i=0$ we identify the set $\Bad(0,1)$ with $\R \times \Bad $ where $\Bad$ is the set of badly approximable numbers. That is, $\Bad(0,1)$ consists of pairs $(\a,\b)$ with $\a \in \R$ and $\b \in \Bad$. If $j=0$ the roles of $\a$ and $\b$ are reversed. We prove that the set $\Bad(1,0)\cap \Bad(0,1) \cap \Bad(i,j)$ has Hausdorff dimension 2, i.e. full dimension. The method easily generalizes to give analogous statements in higher dimensions.

February 15        Mark Watkins (PSU), 10:10, 103 Mcallister:  NOTE earlier time.

Special values of L-functions and modular parametrisations of elliptic curves

Formulae which relate L-values to arithmetic information can be viewed in two directions: you can first compute the arithmetic information to determine the size of the L-value, or conversely you can compute the special L-value by a separate method, thus gaining arithmetic information.  The generic method for computing special L-values (actually any L-value) goes back to Cohen and Zagier in the 1970s, but only recently has it appeared in full generality in print (appendix of Cohen's latest book).  We describe how their method works, and then use it in a specific example, namely the computation of the degree of modular parametrisation of an elliptic curve. In fact, we give data from a large-scale project to compute modular degrees, with over 40000 curves considered.

February 22        Edward Formanek (PSU)

A relation between the Bezoutian and the Jacobian

March 1               Trevor Wooley (University of Michigan) : Colloquium

Slim exceptional sets in Waring's problem

A result of Hua from 1938 shows that the expected asymptotic formula holds in Waring's problem for sums of four squares of primes for almost all integers in the expected residue classes, in the sense that the number of exceptions up to N is O(N(log N)^{-A}). This estimate was recently improved by Liu and Liu to O(N^{13/15+epsilon}). From a naive viewpoint, both conclusions are surprisingly weak, in the sense that a similar conclusion holds already for sums of three squares of primes, and the excess square of a prime brings a negligible improvement in  the estimate for the exceptional set. This phenomenon permeates the subject, especially when the variables under consideration are from such exotic sets as the prime numbers or integers possessing only small prime factors. We present a method for better exploiting excessive variables, especially exotic variables, and thereby slim down the available estimates for associated exceptional sets in various problems of Waring type. By way of illustration, we establish that the above exponent 13/15 may now be replaced by 13/30. As with the best miracle diet plans, this slimming process
involves almost no effort.

March 15             Tonghai Yang  (University of Wisconsin)

Taylor expansion of an Eisenstein series

In this talk, we will give an analogue of the well-known Kronecker limit formula for a classical Eisenstein series (with character). In this case, the Eisenstein series is holomorphic at its center and its central value is given by theta functions via the Siegel-Weil formula. We will give an explicit formula for its central derivative. We will also use the formula to compute the central derivative of certain Hecke L-functions, which are related to CM elliptic curves.

March 22              Joel Anderson (PSU)

March 29              Scott Parsell (Texas A&M)

Pairs of additive equations and inequalities

We will discuss recent progress on obtaining upper bounds for the number of variables required to ensure that a pair of diagonal forms of differing degree, satisfying appropriate local solubility conditions, has a non-trivial integral zero.  The arguments are based on the Hardy-Littlewood method, and in particular on the iterative methods of Vaughan and Wooley for estimating mean values of exponential sums over smooth numbers.  Our estimates can also be applied to the corresponding problem for inequalities, in which one tries to show that two forms with real coefficients assume arbitrarily small values simultaneously at integral points.

April 5                   David Farmer (Bucknell)

Deformation of Maass forms

Phillips and Sarnak conjecture that Maass forms on cofinite subgroups of SL(2,R) are destroyed by almost all deformations of the group.   Some calculations will be described which indicate that "almost all" cannot be replaced by "all."   Time permitting, the dynamics of the motion of the Maass forms under deformation will also be discussed.

April 12                 William Stein (Harvard)

Visibility of Mordell-Weil groups

I will introduce the notions of visibility and modularity of Mordell-Weil groups of abelian varieties.  My notion of visibility is analogous and dual to Barry Mazur's notion of visibility of Shafarevich-Tate groups.  In my talk, I will make conjectures about visibility of Mordell-Weil groups, prove that Mordell-Weil groups of certain elliptic curves are visible in an appropriate restriction of scalars, and give some explicit examples.  If time permits, I will discuss connections with the Birch and Swinnerton-Dyer conjecture for elliptic curves of analytic rank greater than one.

April 19                 Wenzhi Luo (Ohio State University)

Equidistribution of Hecke eigenforms on modular surface

For the holomorphic Hecke eigenforms of weight 2k, one can associate with it naturally a probability measure $mu _{k}$ on the modular surface X. We show  that

         \mu _{k}(A)  = \mu (A) + O(k^{-1/2})

holds uniformly for any set A on X as k tends to infinity, where $\mu$ is the invariant measure associated to the Poincare metric. Moreover the above decay rate is sharp. This equidistribution property of Hecke eigenforms can be regarded as an analogue of ergodicity of Laplacian eigenfunctions.

April 26                  David Boyd (University of British Columbia):  Colloquium

Mahler's measure and the Bloch group