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 F q (t 1/d ) whose members obtain arbitrarily large rank as d → ∞.

No abstract

For any prime , it is possible to construct global function fields whose Jacobians have high -rank by moving to a sufficiently large constant field extension. This was investigated in some detail by Bauer et al. in [2]. The two main results of [2] are an upper bound on the size of the field of definition of the -torsion J [ ] of the Jacobian, and a lower bound on the increase in the base field size that guarantees a strict increase in -rank. Here, we provide improvements to both these results, and give examples which illustrate that our techniques have the potential to yield the correct -rank over any intermediate field of the field of definition of J [ ], including base fields that might be too large to be handled directly by computer algebra packages.

No abstract

No abstract