Explicit equations for Drinfeld modular towers

Abstract

Elaborating on ideas of Elkies, we show how recursive equations for towers of Drinfeld modular curves $(X_0(P^n))_{n\ge 0}$ for $P\in \mathbb F_q[T]$ can be read of directly from the modular polynomial $\Phi_P(X,Y)$ and how this naturally leads to recursions of depth two. Although the modular polynomial $\Phi_T(X,Y)$ is not known in general, using generators and relations given by Schweizer, we find unreduced recursive equations over $\mathbb F_q(T)$ for the tower $(X_0(T^n))_{n\ge 2}$ and of a small variation of it (its partial Galois closure). Reducing at various primes, one obtains towers over finite fields, which are optimal, i.e., reach the Drinfeld--Vladut bound, over a quadratic extension of the finite field. We give a proof of the optimality of these towers, which is elementary and does not rely on their modular interpretation except at one point. We employ the modular interpretation to determine the splitting field of certain polynomials, which are analogues of the Deuring polynomial. For these towers, the particular case of reduction at the prime $T-1$ corresponds to towers introduced by Elkies and Garcia--Stichtenoth.