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8. Discriminants

    1. distribution of discriminants


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      Problem 8.1.

      [Ian Whitehead] This is a question suggested by Jordan Ellenberg at Arizona Winter School several years ago. Consider all monic polynomials over $\mathbb F_q$ all of fixed degree $d$. What is the distribution of discriminants? Here, we would like to compute the distribution for fixed $q$ and $d$. Whitehead says he knows how to do this via analytic techniques (even though it is not written down) and it could be interesting to understand things geometrically via understanding the homology of certain relevant spaces.

      Phil Tosteson points out the cohomology of the relevant space has been computed by Edward Frenkel, who computed the cohomology of the commutator subgroup of the braid group. One may still have to figure out the relevant Frobenius action on the cohomology. It was clear that some of the cohomology is not of Tate type, but it has something to do with CM elliptic curves.

          Cite this as: AimPL: Moments in families of L-functions over function fields, available at http://aimpl.org/momentsoverff.