Cohen Lenstra Heuristics for \'Etale Group Schemes and Symplectic Pairings

We generalize the Cohen-Lenstra heuristics over function fields to \'{e}tale 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 $\wedge^2G(1)$, 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 Garton, EVW, 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.