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

Explicit equations for Drinfeld modular towers