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In this article, we give a conjecture for the average number of unramified -extensions of a quadratic field for any finite group . The Cohen–Lenstra heuristics are the specialization of our conjecture to the case in which is abelian of odd order. We prove a theorem toward the function field analogue of our conjecture and give additional motivations for the conjecture, including the construction of a lifting invariant for the unramified -extensions that takes the same number of values as the predicted average and an argument using the Malle–Bhargava principle. We note that, for even , corrections for the roots of unity in are required, which cannot be seen when is abelian.
Given a polynomial function with an isolated zero at the origin, we prove that the local -Brouwer degree equals the Eisenbud–Khimshiashvili–Levine class. This answers a question posed by David Eisenbud in 1978. We give an application to counting nodes, together with associated arithmetic information, by enriching Milnor’s equality between the local degree of the gradient and the number of nodes into which a hypersurface singularity bifurcates to an equality in the Grothendieck–Witt group.
Using a representation of the discrete Hilbert transform in terms of martingales arising from Doob -processes, we prove that its -norm, , is bounded above by the -norm of the continuous Hilbert transform. Together with the already known lower bound, this resolves the long-standing conjecture that the norms of these operators are equal.