A survey of Hurwitz spaces in inverse Galois theory that reviews geometric and arithmetic developments and highlights recent constructions of rational components plus applications to heuristics over finite fields.
Homological stability for Hurwitz spaces and the Cohen-Lenstra conjecture over function fields, II
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We prove a version of the Cohen--Lenstra conjecture over function fields (completing the results of our prior paper). This is deduced from two more general theorems, one topological, one arithmetic: We compute the direct limit of homology, over puncture-stabilization, of spaces of maps from a punctured manifold to a fixed target; and we compute the Galois action on the set of stable components of Hurwitz schemes.
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Hurwitz spaces and Inverse Galois Theory
A survey of Hurwitz spaces in inverse Galois theory that reviews geometric and arithmetic developments and highlights recent constructions of rational components plus applications to heuristics over finite fields.