Towers of function fields and iterated means

Towers of function fields meeting the Drinfeld-Vladut bound with equality were constructed by Gorcal-Stichtenoth (1995, 1996). Modular uniformizations thereof were pointed out by Elkies (1996). We connect this fact to hypergeometric analogs of the arithmetic-geometric mean and related theta series identities discovered by Borweins and Garven.