Intermediate extension in arithmetic statistics

Will Sawin, Princeton University
Fine Hall 322

Davenport-Heilbronn and Bhargava counted cubic, quartic, and quintic fields by first counting a larger set of orbits in some prehomogenous vector space and then sieving out the orbits corresponding to fields. Several mathematicians have studied moments of Dirichlet L-functions by first estimating multiple Dirichlet series which are sums of Dirichlet L-functions times auxiliary terms and then sieving out the moments of L-functions. Recent results of myself and my student Kevin Chang, building on ideas of Diaconu-Pasol, Ellenberg-Tran-Westerland, and Kapranov-Schechtman, demonstrate that both of these strategies, in the function field context, have the same geometric interpretation: First sum the Frobenius trace function of the intermediate extension of some sheaf of interest and then sieve out the sum of the trace function of the original sheaf. I will explain this interpretation and speculate on its consequences.