Title: Parity in Iwasawa Theory for CM Elliptic Curves
Abstract: A fundamental invariant of an elliptic curve E over a number
field K is the rank of the group E(K) of K-rational points
of E. The growth of the rank of E(L) as L varies over
certain extensions of K of p-power order can be described in
terms of constants, called Iwasawa invariants, attached to
E. In many cases, work of Nekovar and Mazur-Rubin allows one
to identify the Iwasawa invariants of E modulo 2. In this
talk, we present similar resuls for Iwasawa invariants
modulo 4 in the case when E has complex multiplication.
Title: Several Problems of Littlewood (and others)
Abstract: A number of classical and not so classical problems in number
theory concern finding polynomials with integer coefficients that are
small in some way. Typically these problems lie somewhere between
Diophantine Number Theory, Harmonic Analysis and Combinatorics
These include old chestnuts like the Merit Factor Problem of Golay,
Lehmer's Conjecture and various Littlewood Conjectures.
Title: Explicit Modular Families of K3 Surfaces
Abstract: K3 surfaces are one natural generalization of elliptic curves to a class of algebraic surfaces, others being abelian surfaces (complex 2-dimensional tori with polarization) or elliptic surfaces (relative elliptic curves over a base curve). The notion of "normal form" for an elliptic curve (e.g., Weierstrass, Hesse, or Legendre normal forms) can be adapted to two classes of K3 surfaces whose moduli spaces are naturally identified with a Hilbert modular surface or Siegel modular threefold. Motivated by both the Hodge conjecture and ideas from string theory -- background in neither case being assumed -- we will explicitly describe algebraic correspondences between these two special families of K3 surfaces and associated families of abelian surfaces. This will be accomplished through special realizations of the K3 surfaces as elliptic surfaces, so that all the natural algebraic
surface generalizations of elliptic curves are seen to be interrelated. This is joint work with Adrian Clingher.
Title: Small Gaps between Primes
Abstract: I will discuss some of the ideas that are used in the recent result of Goldston-Pintz-Yildirim which shows that there are always primes much closer together than the average distance between consecutive primes. The method depends on the distribution of primes in arithmetic progressions, and subject to an unproved conjecture on this distribution the method even produces infinitely many pairs of primes whose difference is 16 or less. If times allows I may show some of the media coverage this result generated a few years ago.
Title: Lattice Points, Toric Varieties, and Zeta Functions
Abstract: The problem of giving exact formulas for the number of lattice
points in a convex polytope has interested mathematicians for many
years. Pick's Formula (c. 1890) gives the answer in dimension two, and
Ehrhart achieved interesting results in higher dimensions in the
1960's. In the past fifteen years, much progress has been made using
the tool of toric varieties. Recent toric results of Brion, Morelli,
Khovanskii, and the speaker have helped us understand the lattice
point question much more clearly. In addition, there is a close link
between the number theory which arises in these formulas (Dedekind
sums and their generalizations) and special values of zeta functions.
In particular, this link provides a new conceptual understanding of
formulas of Shintani and Zagier (1970's) which express values of zeta
functions of real quadratic fields at negative integers.
Title: A p-Adic Construction of Global Points on Elliptic Curves over Imaginary Quadratic Fields
Abstract: An elliptic curve E over an imaginary quadratic field F is in
most cases conjectured to correspond to a weight 2 cusp form on GL_2(AF)
(a la Shimura-Taniyama). Such forms admit an elementary description as
harmonic differentials on quotients of the upper half-space, and are the
only type of modular form other than on GL2(Q) to possess an analogous,
1-dimensional modular symbol. Using this modular symbol we construct
certain measures on P1(Cp), p a characteristic of bad reduction, and
define period integrals whose image under the Tate parametrization
conjecturally yields points defined over class fields of a suitable
quadratic extension K/F, thus giving an answer to Hilbert's 12th problem