The 3rd MSJ-SI

The Mathematical Society of Japan, Seasonal Institute

Development of Galois-Teichmüller Theory
and Anabelian Geometry

October 26 Tuesday, 17:10--18:00
J. Ellenberg (University of Wisconsin)
Ihara's braid group and fundamental groups of random curves
Let p be a prime and S a finite set of primes not including p. Let G_S(p) be the Galois group of the maximal pro-p extension of Q unramified away from S. What does G_S(p) look like when S is a "random" set of primes of fixed size? Questions of this kind pertaining to _abelian_ unramified pro-p extensions of number fields (i.e. p-parts of ideal class groups) are the subject of the Cohen-Lenstra conjectures. But the non-abelian case has been studied much less. We discuss two routes to a heuristic for the distribution of G_S(p); one along the lines of the original Cohen-Lenstra argument, and another via the analogy with function fields, in which we model the action of Frobenius on the arithmetic fundamental group of a curve by a random element of Ihara's pro-p braid group. It turns out that both routes lead to the same heuristic, which agrees with the few results one can prove, and is reasonably consistent with the experimental data we can gather. This is join!
Work with Nigel Boston -- a preprint can be seen at