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Cohen–Lenstra heuristics for étale group schemes and symplectic pairings
- Part of
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- Journal:
- Compositio Mathematica / Volume 155 / Issue 4 / April 2019
- Published online by Cambridge University Press:
- 20 March 2019, pp. 758-775
- Print publication:
- April 2019
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- Article
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We generalize the Cohen–Lenstra heuristics over function fields to étale 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 
$\wedge ^{2}G(1)$, we formulate more refined heuristics which nicely explain the deviation from the usual Cohen–Lenstra heuristics for abelian 
$\ell$-groups in cases where 
$\ell \mid q-1$; 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.
