On the Deuring Polynomial for Drinfeld Modules in Legendre Form

We study a family $\psi^{\lambda}$ of $\mathbb F_q[T]$-Drinfeld modules, which is a natural analog of Legendre elliptic curves. We then find a surprising recurrence giving the corresponding Deuring polynomial $H_{p(T)}(\lambda)$ characterising supersingular Legendre Drinfeld modules $\psi^{\lambda}$ in characteristic $p(T)$.Comment: This article supersedes arXiv:1110.607

A new tower over cubic finite fields

We present a new explicit tower of function fields (Fn)n≥0 over the finite field with ` = q3 elements, where the limit of the ratios (number of rational places of Fn)/(genus of Fn) is bigger or equal to 2(q2 − 1)/(q + 2). This tower contains as a subtower the tower which was introduced by Bezerra– Garcia–Stichtenoth (see [3]), and in the particular case q = 2 it coincides with the tower of van der Geer–van der Vlugt (see [12]). Many features of the new tower are very similar to those of the optimal wild tower in [8] over the quadratic field Fq2 (whose modularity was shown in [6] by Elkies).

A complete characterization of Galois subfields of the generalized Giulietti--Korchm\'aros function field

We give a complete characterization of all Galois subfields of the generalized Giulietti--Korchm\'aros function fields \mathcal C_n / \fqn for $n\ge 5$. Calculating the genera of the corresponding fixed fields, we find new additions to the list of known genera of maximal function fields

Good families of Drinfeld modular curves

In this paper we investigate examples of good and optimal Drinfeld modular towers of function fields. Surprisingly, the optimality of these towers has not been investigated in full detail in the literature. We also give an algorithmic approach on how to obtain explicit defining equations for some of these towers and in particular give a new explicit example of an optimal tower over a quadratic finite field

Towers of Function Fields over Non-prime Finite Fields

Over all non-prime finite fields, we construct some recursive towers of function fields with many rational places. Thus we obtain a substantial improvement on all known lower bounds for Ihara's quantity $A(\ell)$, for $\ell = p^n$ with $p$ prime and $n>3$ odd. We relate the explicit equations to Drinfeld modular varieties

Towers of function fields over cubic fields

Bezerra, Garcia and Stichtenoth constructed an explicit tower of function fields over a cubic finite field, whose limit attains the Zink bound. Their proof is rather long and very technical. The main aim of this thesis is to replace the complex calculations in their work by structural arguments, thus giving a much simpler and more transparent proof for the limit of the Bezerra–Garcia–Stichtenoth tower. We also compute the limit of the Galois closure of this tower. One of the main tools used while determining the limits of these towers is a lemma from ramification theory. Using the theory of higher ramification groups, we give proof of this result, which is valid for more general fields. Furthermore, using a variant of these towers, we obtain asymptotic lower bounds for the class of r-quasi transitive codes over cubic finite fields and the class of transitive isoorthogonal codes over cubic finite fields