Modular Towers Construction and Diophantine Questions

Modular towers, a notion due to M. Fried, are towers of Hurwitz spaces, with levels corresponding to the characteristic quotients of the p-universal Frattini cover of a fixed finite group G (with p a prime divisor of |G|). The tower of modular curves X1(pn) (n>0) is the original example: the group G is then the dihedral group Dp. There are diophantine conjectures on modular towers, inspired by modular curves: the spirit is that over a number field, rational points do not exist beyond a certain level. In this paper, which is the first of a series of three on this topic in this volume, after defining modular towers, we discuss the significance of these conjectures and explain some known results.