Modular Towers of Noncongruence Curves

This proposal extends much work done during the last funding period. Two topics, however, which have separate goals, have brief summaries in x10. Together they form documentation of the results that came from the funding period of NSF GRANT #9622928. 1. An outline of the main problems of this proposal Let G be a nite group. Call a conjugacy class C a p 0-class if its elements have order prime to p. Modular Towers is the name I've given to a sequence of moduli spaces canonically attached to a simple set of data coming from G. This data is a prime p dividing jGj and a collection of p 0-classes C = C 1 The kth level is the manifold H rd k = H(G k ; C) rd that x3.2 explains. Here J r is the moduli space of r distinct unordered points. In particular, J 4 is the classical j-line minus the point at innnity. To simplify notation, multiply j by a constant to assume 0 (resp. 1) corresponds to the isomorphism class of elliptic curves with six (resp. 4) automorphisms xing the origin. Denote the absolute Galois group of a eld K by G K and the Grothendieck-Teichm uller group by c GT. This section describes the scope of the investigations in this proposal. Later sections provide evidence there has been previous accomplishment and we are ready to tackle these problems. UCI graduate students Paul Bailey and Darren Semmen work with me, respectively, on computations from x4 and x5. I hope to get support for them, too. Modular Towers applies modular curve type thinking to many applications not previously associated with such techniques. The sequence (1.1) has a simple definition , in ways simpler than for modular curves. The latter is the case G = D p (the dihedral group of order 2p; p an odd prime) and C is four repetitions of the involution conjugacy class. (Then, H rd k is Y 1 (p k+1) whose projective completion is X 1 (p k+1), k = 0; 1; : : : .) Further, Modular Towers has built-in interpretations of the Inverse Galois Problem. This proposal illustrates how these two aspects help adapt Modular Towers to classical problems, those recognized by researchers in modular curves and in hyperbolic geometry (interested in compelling proonite groups like c GT).