Local-global problem for Drinfeld modules

Let K be a function field with an A-algebra structure. The ring A arises in the definition of the Drinfeld module φ over K. By E(K) we denote K together with the A-module structure induced on it by φ. For any principal prime ideal (a)⊂A, we study the question whether an element x∈E(K) which is an a-fold in E(K_ν) for every place ν of K, is an a-fold in E(K). In particular, we study the group

     

On a conjecture of Whittaker concerning uniformization of hyperelliptic curves

This article concerns an old conjecture due to E. T. Whittaker, aiming to describe the group uniformizing an arbitrary hyperelliptic Riemann surface as an index two subgroup of the monodromy group of an explicit second order linear differential equation with singularities at the values .

Whittaker and collaborators in the thirties, and R. Rankin some twenty years later, were able to prove the conjecture for several families of hyperelliptic surfaces, characterized by the fact that they admit a large group of symmetries. However, general results of the analytic theory of moduli of Riemann surfaces, developed later, imply that Whittaker's conjecture cannot be true in its full generality.

Recently, numerical computations have shown that Whittaker's prediction is incorrect for random surfaces, and in fact it has been conjectured that it only holds for the known cases of surfaces with a large group of automorphisms.

The main goal of this paper is to prove that having many automorphisms is not a necessary condition for a surface to satisfy Whittaker's conjecture.

     

Covariant Hyperelliptic Functions of Genus Two

We report results of investigations concerning the role of representations of in the theory of genus-two hyperelliptic functions. We discuss the role of these representations in the classical theory as well as introduce a family of new, naturally covariant functions.     

Helly-type theorems for homothets of planar convex curves

Helly's theorem implies that if is a finite collection of (positive) homothets of a planar convex body , any three having non-empty intersection, then has non-empty intersection. We show that for collections of homothets (including translates) of the boundary , if any four curves in have non-empty intersection, then has non-empty intersection. We prove the following dual version: If any four points of a finite set in the plane can be covered by a translate [homothet] of , then can be covered by a translate [homothet] of . These results are best possible in general.

     

Iwasawa μ-invariants of elliptic curves and their symmetric powers

In an earlier paper we considered the effects that finite submodules can have on μ-invariants of Selmer groups. In this paper we examine some of the consequences of that theory to elliptic curves and their symmetric powers. One main result is the construction of an isogeny class of elliptic curves, all of which have positive μ-invariants. A second result is a connection between the behaviors of μ-invariants associated with the symmetric powers of an elliptic curve over , and the behavior of the μ-invariant of that elliptic curve over different extensions of .     

Higher Weierstrass points on

We study the arithmetic properties of higher Weierstrass points on modular curves for primes . In particular, for , we obtain a relationship between the reductions modulo of the collection of -Weierstrass points on and the supersingular locus in characteristic .

     

Taylor expansion of an Eisenstein series

In this paper, we give an explicit formula for the first two terms of the Taylor expansion of a classical Eisenstein series of weight for . Both the first term and the second term have interesting arithmetic interpretations. We apply the result to compute the central derivative of some Hecke -functions.

     

Sieving for rational points on hyperelliptic curves

We give a new and efficient method of sieving for rational points on hyperelliptic curves. This method is often successful in proving that a given hyperelliptic curve, suspected to have no rational points, does in fact have no rational points; we have often found this to be the case even when our curve has points over all localizations . We illustrate the practicality of the method with some examples of hyperelliptic curves of genus .

     

A Note on Shiffman's Theorems

Shiffman proved his famous first theorem, that if A R³ is a compact minimal annulus bounded by two convex Jordan curves in parallel (say horizontal) planes, then A is foliated by strictly convex horizontal Jordan curves. In this article we use Perron's method to construct minimal annuli which have a planar end and are bounded by two convex Jordan curves in horizontal planes, but the horizontal level sets of the surfaces are not all convex Jordan curves or straight lines. These surfaces show that unlike his second and third theorems, Shiffman's first theorem is not generalizable without further qualification.