The Galois closure of Drinfeld modular towers

In this article we study Drinfeld modular curves X₀(pn) associated to congruence subgroups Γ₀(pn) of GL(2,Fq[T]) where p is a prime of Fq[T]. For n>r>0 we compute the extension degrees and investigate the structure of the Galois closures of the covers X₀(pn)→X₀(pr) and some of their variations. The results have some immediate implications for the Galois closures of two well-known optimal wild towers of function fields over finite fields introduced by Garcia and Stichtenoth, for which the modular interpretation was given by Elkies.