The `-Rank Structure of a Global Function Field

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.