We prove a homological stabilization theorem for Hurwitz spaces: modulispaces of branched covers of the complex projective line. This has thefollowing arithmetic consequence: let l>2 be prime and A a finite abelianl-group. Then there exists Q = Q(A) such that, for q greater than Q and notcongruent to 1 modulo l, a positive fraction of quadratic extensions of F_q(t)have the l-part of their class group isomorphic to A.
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