Let \(\mathbb{K}/\mathbb{F}_q\) be a function field of genus \(g,\) and \(\ell\) a prime not dividing \(q.\) It is known that the Jacobian of \(\mathbb{K}/\mathbb{F}_q\) has \(\ell\)-rank \(2g\) over \(\overline{\mathbb{F}}_q.\) One question of interest is to find \(n_\ell \in \mathbb{Z}^+\) such that the Jacobian of \(\mathbb{K}/\mathbb{F}_{q^{n_\ell}}\) has full \(\ell\)-rank. A second is to find \(n \in \mathbb{Z}^+\) such that the Jacobian of \(\mathbb{K}/\mathbb{F}_{q^n}\) has \(\ell\)-rank larger than that over \(\mathbb{F}_q,\) and to quantify exactly how much larger. These questions were both partially addressed by M. Bauer et al. [“Construction of hyperelliptic function fields of high three-rank”, Math. Comput. 77, No. 261, 503–530 (2008; Zbl 1131.11073)] by using information obtained from the factorization modulo \(\ell\) of the reciprocal polynomial of the \(L\)-polynomial of \(\mathbb{K}/\mathbb{F}_q\) and some linear algebra computations. This paper improves and extends these results. It is shown that the upper bound on \(n_\ell\) obtained by Bauer et al. can be computed more easily, in particular reducing the number of matrix order computations required, and that it is in fact a multiple of \(n_\ell.\) Another method of computing an upper bound that does not require any matrix order computations is also presented. Finally, the result of Bauer et al. on incrementally increasing the \(\ell\) rank is also improved; for every \(n\) between \(1\) and \(n_\ell,\) a bound on the \(\ell\)-rank of \(\mathbb{K}/\mathbb{F}_{q^n}\) can be computed that is, in many cases, exactly equal to the \(\ell\)-rank. This result again only requires the \(L\)-polynomial of \(\mathbb{K}/\mathbb{F}_{q^n}\) and the computation of some matrix orders.
For the entire collection see [Zbl 1210.11005].