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Random braids, finite extensions of global fields, stable cohomology, and variations on Cohen-Lenstra

Ellenberg, J (Wisconsin-Madison)
Wednesday 26 August 2009, 11:30-12:30

Seminar Room 1, Newton Institute


It is by now a common technique to construct and evaluate conjectures about number fields by means of analogous conjectures over function fields. For example, the Cohen-Lenstra conjecture about the statistics of ell-parts of class groups of quadratic imaginary fields can be thought of, on the function field side, as a conjecture about the statistics of the finite group Jac(X)[ell^infty](F_q), where X is a "random" hyperelliptic curve over F_q of large genus. In this setting, the Cohen-Lenstra conjecture is compatible with the heuristic that the image of Frobenius is a random symplectic matrix in Aut(T_ell Jac(X)). I will argue, somewhat against the grain of this conference, that in this setting the anabelian story may not be so different from the abelian story. In particular, one may productively study the statistics of non-abelian extensions of global fields by means of the heuristic that the action of Frobenius on pi_1^{et}(X/F_q) should act as a random element of (an appropriate subgroup of) Ihara's profinite braid group. To make this more concrete, I will explain a) how to prove a weak version of the Cohen-Lenstra conjecture over F_q(t) via a new theorem on stable cohomology of Hurwitz spaces (joint work with A. Venkatesh, C. Westerland) and b) how to frame a "non-abelian pro-p Cohen-Lenstra conjecture," describing the statistics of the Galois group of the maximal pro-p extension unramified away from a random set of primes ell_1, .. ell_r congruent to 1 mod p. (joint work with N. Boston, A. Venkatesh.)


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