Ranks of abelian varieties in towers of function fields

Abstract

With the goal of producing elliptic curves and higher dimensional abelian varieties of large rank over function fields, we provide a geometric construction of towers of surfaces dominated by products of curves; in the case where the surface is defined over a finite field our construction yields families of smooth, projective curves whose Jacobians satisfy the conjecture of Birch and Swinnerton-Dyer. As an immediate application of our work we employ known results on analytic ranks of abelian varieties defined in towers of function field extensions, producing a one-parameter family of elliptic curves over [special characters omitted](t^1/d) whose members obtain arbitrarily large rank as d → ∞.