Elliptic Curves with Bounded Ranks in Function Field Towers

We study the arithmetic structure of elliptic curves over k(t), where k is an algebraically closed field. In [Shi86] Shioda shows how one may determine rank of the Néron-Severi group of a Delsarte surface–a surface that may be defined by four monomial terms. To this end, he describes an explicit method of computing the Lefschetz number of a Delsarte surface. He proves the universal bound of 56 on the rank of an elliptic curve defined by an equation of the form y2 = x3 + atnx+ btm over k(t), where k is an algebraically closed field of characteristic zero. In [Shi92] Shioda shows that the rank of 68 is obtained for the curve y2 = x3 + t360 + 1 over C(t). In recent work, Heinje [Hei11] characterizes all Delsarte elliptic surfaces. He determines 42 families of Delsarte elliptic curves and shows, through explicit computation, that 68 is the maximal rank over k(t), k algebraically closed of characteristic zero. By relating a Delsarte surface to a Fermat surface, Shioda is able to exploit the relationship between divisor classes on his surface and the Mordell-Weil group of its generic fiber. In [Ber08] the author describes a more flexible construction of elliptic surfaces. We explicitly construct families of surfaces, dominated by products of curves, with the additional property that they retain this DPC property under base extension. The Néron Severi group of a product of curves may be expressed in terms of divisorial correspondences on the product, and Ulmer [Ulm11] utilizes this relationship to prove an explicit formula for the ranks of the Jacobians of the curves constructed in [Ber08]. He produces elliptic curves with rank at least 13 over C(t), and Occhipinti [Occ10] produces an elliptic curve over F̄p(t) whose ranks over the fields F̄p(t) grow at least linearly with d prime to p. The goal of this note is to show that the large rank examples obtained via our construction are rare. We determine all elliptic curves obtained via the construction in [Ber08], and we find that, for all but finitely many families, the Mordell-Weil group of E/k(t1/d) has rank zero, for each d prime to the characteristic of K = k(t), k an algebraically closed field of arbitrary characteristic.