Introduction to modular towers: Generalizing dihedral group-modular curve connections. (English) Zbl 0957.11047

Fried, Michael D. (ed.) et al., Recent developments in the inverse Galois problem. A joint summer research conference, July 17-23, 1993, University of Washington, Seattle, WA, USA. Providence, RI: American Mathematical Society. Contemp. Math. 186, 111-171 (1995).
Summary: We join Hurwitz space constructions and the universal Frattini cover of a finite group. The goal is to form and apply generalizations of the towers \(X_0(p) \leftarrow X_0(p^2) \cdots\leftarrow X_0(p^n) \leftarrow \cdots\) of modular curves. This generalization relies on the appearance of the dihedral group \(D_p\) and its companion group \(\mathbb{Z}_p \times^s \{\pm 1\}\) in the theory of modular curves. We replace \(D_p\) by any finite group \(G\), and \(p\) by any prime dividing the order of \(G\). The replacement for \(\mathbb{Z}_p \times^s \{\pm 1\}\) is the Universal \(p\)-Frattini Cover of \(G\).
Diophantine motivations include an outline for using the Ihara-Drinfeld relations for the Grothendieck-Teichmüller group. Conjecturally this is the absolute Galois group of \(\mathbb{Q}\). We consider how finding fields of definition of absolutely irreducible components of Hurwitz spaces can test this. There are many applications to finite fields. The simplest bounds the exceptional primes to realizing any finite group \(G\) as the Galois group of a regular extension of \(\mathbb{F}_p(x)\). The structure of Modular Towers connects this problem to other diophantine problems.
Alternating groups test the modular representation theory that appears in Part II of the paper. These give a modular tower different from modular curves. The classical link here is to theta functions with characteristic.
For the entire collection see [Zbl 0823.00012].


11R32 Galois theory
14G32 Universal profinite groups (relationship to moduli spaces, projective and moduli towers, Galois theory)
11G18 Arithmetic aspects of modular and Shimura varieties
11R58 Arithmetic theory of algebraic function fields