Explicit equations for Drinfeld modular towers

TL;DR

This paper derives explicit recursive equations for Drinfeld modular towers from modular polynomials, constructs optimal towers over finite fields, and proves their optimality with minimal reliance on modular interpretation.

Contribution

It provides a method to obtain recursive equations for Drinfeld modular towers directly from modular polynomials, including new towers over finite fields that are proven to be optimal.

Findings

Recursive equations for towers derived from modular polynomials.

Construction of towers reaching the Drinfeld--Vladut bound.

Elementary proof of the towers' optimality.

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.