Jacobi Sums, Fermat Jacobians, and Ranks of Abelian Varieties over Towers of Function Fields

1.1. Given an abelian variety A over a function field K = k(C) with C an absolutely irreducible, smooth, proper curve over a field k, it is natural to ask about the behavior of the Mordell-Weil group of A in the layers of a tower of fields over K. The simplest case, which is already very interesting, is when A is an elliptic curve, K = k(t) is a rational function field, and one considers the towers k(t) or k(t) as d varies through powers of a prime or through all integers not divisible by the characteristic of k. When k = Q or more generally a number field, several authors (e.g., [Shi86], [Sti87], [Fas97], [Sil00], [Sil04], and [Ellen]) have considered this question and given bounds on the rank of A over Q(t) or Q(t). In some interesting cases it can be shown that A has rank bounded independently of d in the tower Q(t). Of course no example is yet known of an elliptic curve over Q(t) with unbounded ranks in the tower Q(t), nor of an elliptic curve over Q(t) with non-constant j-invariant and unbounded ranks in the tower Q(t). When k is a finite field, examples of Shioda and the author show that there are non-isotrivial elliptic curves over Fp(t) with unbounded ranks in the towers Fp(t) [Shi86, Remark 10] and Fp(t) [Ulm02, 1.5]. More recently, the author has shown [Ulmer] that high ranks over function fields over finite fields are in some sense ubiquitous. For example, for every prime p and every integer g > 0 there are absolutely simple abelian varieties of dimension g over Fp(t) with unbounded ranks in the tower Fp(t), and given any non-isotrivial elliptic curve E over Fq(t), there exists a finite extension Fr(u) such that E has unbounded (analytic) ranks in the tower Fr(u). One obvious difference between number fields and finite fields which might be relevant here is the complexity of their absolute Galois groups: that of a finite field is pro-cyclic while that of a number field is highly non-abelian. Ellenberg uses this non-abelianess in a serious way in his work on bounding ranks and, in a private communication, he asked whether it might be the case that, say, a non-isotrivial elliptic curve over Fq(t) always has unbounded rank in the tower Fp(t). Our goal in this note, which is a companion to [Ulmer], is to give a number of examples of abelian varieties over function fields Fq(t) which have bounded ranks