Douglas Ulmer

Elliptic curves with large rank over function fields

We produce explicit elliptic curves over Fp(t) whose Mordell-Weil groups have arbitrarily large rank. Our method is to prove the conjecture of Birch and Swinnerton-Dyer for these curves (or rather the Tate conjecture for related elliptic surfaces) and then use zeta functions to determine the rank. In contrast to earlier examples of Shafarevitch and Tate, our curves are not isotrivial. Asymptotically these curves have maximal rank for their conductor. Motivated by this fact, we make a conjecture about the growth of ranks of elliptic

Explicit points on the Legendre curve

Abstract We study the elliptic curve E given by y 2 = x ( x + 1 ) ( x + t ) over the rational function field k ( t ) and its extensions K d = k ( μ d , t 1 / d ) . When k is finite of characteristic p and d = p f + 1 , we write down explicit points on E and show by elementary arguments that they generate a subgroup V d of rank d − 2 and of finite index in E ( K d ) . Using more sophisticated methods, we then show that the Birch and Swinnerton-Dyer conjecture holds for E over K d , and we relate the index of V d in E ( K d ) to the order of the Tate–Shafarevich group Ш ( E / K d ) . When k has characteristic 0, we show that E has rank 0 over K d for all d .

L-functions with large analytic rank and abelian varieties with large algebraic rank over function fields

The goal of this paper is to explain how a simple but apparently new fact of linear algebra together with the cohomological interpretation of L-functions allows one to produce many examples of L-functions over function fields vanishing to high order at the center point of their functional equation. The main application is that for every prime p and every integer g>0 there are absolutely simple abelian varieties of dimension g over Fp(t) for which the BSD conjecture holds and which have arbitrarily large rank.

Elliptic Curves and Analogies Between Number Fields and Function Fields

Well-known analogies between number fields and function fields have led to the transposition of many problems from one domain to the other. In this paper, we discuss traffic of this sort, in both directions, in the theory of elliptic curves. In the first part of the paper, we consider various works on Heegner points and Gross–Zagier formulas in the function field context; these works lead to a complete proof of the conjecture of Birch and Swinnerton-Dyer for elliptic curves of analytic rank at most 1 over function fields of characteristic > 3. In the second part of the paper, we review the fact that the rank conjecture for elliptic curves over function fields is now known to be true, and that the curves which prove this have asymptotically maximal rank for their conductors. The fact that these curves meet rank bounds suggests interesting problems on elliptic curves over number fields, cyclotomic fields, and function fields over number fields. These problems are discussed in the last four sections of the paper.

On Mordell–Weil groups of Jacobians over function fields

We study the arithmetic of abelian varieties over

Geometric non-vanishing

K=k(t)

On universal elliptic curves over Igusa curves.

where

Explicit points on the Legendre curve III

k

Curves and Jacobians over Function Fields

is an arbitrary field. The main result relates Mordell-Weil groups of certain Jacobians over

L-Functions of Universal Elliptic Curves Over Igusa Curves

K