< style="color: rgb(102, 0, 204);"> Arithmetic Galois Theory and Related Moduli Spaces
< style="color: rgb(102, 0, 204);">R.I.M.S., Kyoto, October 23-27 & 30-31, 2006
l-adic aspects of the Modular Tower program

Pierre D`ebes

Universit´e Lille 1

<> Abstract: There is an obstruction to realizing a finite G as a regular Galois group over Q  with a bounded number of branch points: if G has a p-subgroup with a big abelianization, then branch points should coalesce modulo small primes or p should divide some ramification index. We will explain this and will give some consequences for the Regular Inverse Galois problem and for the Modular Tower program where the obstruction comes in full light and leads to some weak version of the main conjecture of this program. We will also recast previous results on l - adic realizations in this perspective.

References:

P. Bailey and M. Fried, Hurwitz monodromy, spin separation and higher levels of a Modular Tower, Proceedings of Symposia in Pure Mathematics, 70, AMS, ed. by M. Fried and

Y. Ihara, (2002), 70–220.

A. Cadoret, Modular Towers and Torsion on Abelian Varieties, preprint.

P. D`ebes and M. Emsalem, Harbater-Mumford Components and Towers of Moduli Spaces,

J. Math. Inst. Jussieu, 5/03, (2006), 351–371.