The ℓ-rank structure of a global function field

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.