Incomplete Saturday, March  1, 2008: A Moduli Approach to Generalizing Serre's Open Image Theorem

Results of Vasiu and Voelklein and mine, mesh together to give hints that for (what I call) abelianized Modular Towers if you go up high enough in the tower, you get relatively Frattini. For example, even in the bad primes p=2 and p=3 for Serre's result, you get this for the modular curve case (by the second level). Further, in what I call higher rank towers (where all but a finite set of primes come into play and you conjecturally get moduli-defined "Hecke" operators) you expect all the tower levels to have monodromy over the base that is a p-Frattini cover of the level 0 monodromy for almost all primes p. Again this is the little lemma of Serre that PSL_2(\bZ/p^{k+1})\to PSL_2(\bZ/p) is a p-Frattini cover for all k\ge 0, and p\not 2 or 3 in the modular curve case. 

While this is just the first step in getting an Open Image Theorem, the proof I have of this in cases is based on something deeper. That is, my approach to the more explicit way I get a hold of cusps in Modular Towers. Even when the tower levels identify with Shimura varieties, this is a different approach to cusps. The main results (using a Fried-Serre result) show the existence of (sufficiently many of) what I call p-cusps. That is the paper I'm finishing now -- I've put up the the listing of it below (where the phrase "Atomic Orbital Type" occurs; I'm still finishing the applications in that, though). 

So, in my best cases I'm able to show the second step of Serre's OIT: That for "j-line" values p-adically "close to" \infty, the OIT holds. I've also set up one example (nonmodular curve case) the \S 6 of the second listing from my web site below, called the Z/3 case (versus the Z/2 case that is modular curves), for which my conjectures mesh nicely with actual literature from someone (Gabriel Berger) who worked with me as a post-doctoral.   

The advantage of the Modular Tower approach over Serre's (mostly unpublished) attempt at "generic abelian varieties" approach is that mine, using a moduli approach based on finite group theory doesn't expect to need a "generic" type statement. Everything in the paper the Luminy paper below states the conjectures as based on Frattini principles. This is not a minor point, though I stop discussing it here.