Cohen-Lenstra heuristics for etale group schemes and symplectic pairings

摘要

We generalize the Cohen-Lenstra heuristics over function fields to etale group schemes G (with the classical case of abelian groups corresponding to constant group schemes). By using the results of Ellenberg-Venkatesh-Westerland, we make progress towards the proof of these heuristics. Moreover, by keeping track of the image of the Weil-pairing as an element of Lambda(2)G (1), we formulate more re fined heuristics which nicely explain the deviation from the usual Cohen-Lenstra heuristics for abelian l-groups in cases where l vertical bar q - 1; the nature of this failure was suggested already in the works of Malle, Garton, Ellenberg-Venkatesh-Westerland, and others. On the purely large random matrix side, we provide a natural model which has the correct moments, and we conjecture that these moments uniquely determine a limiting probability measure.