BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Sean Howe (University of Utah) DTSTART:20250204T210000Z DTEND:20250204T220000Z DTSTAMP:20260423T130421Z UID:MITNT/108 DESCRIPTION:Title: Sideways equidistribution of function field L-functions\nby Sean Howe (University of Utah) as part of MIT number theory seminar\n\nLecture held in Room 2-449 in the Simons Building (building 2).\n\nAbstract\nIn the fir st part of this talk\, we will explain a concise description of the asympt otic distributions of eigenvalues of Haar-random orthogonal matrices using a new $\\sigma$-moment generating function that replaces the usual expone ntial with the plethystic exponential of symmetric function theory. Simila r descriptions can be obtained also for compact symplectic\, unitary\, and symmetric groups. \n\nIn the second part of the talk\, we will explain ho w to use point-counting techniques to compute\, for a fixed finite field $ \\mathbb{F}_q$\, the distribution of the zeroes of the $L$-function of a r andom smooth degree $d$ surface in $\\mathbb{P}^3_{\\mathbb{F}_q}$ as $d \ \rightarrow \\infty$. The result is a simple description of the asymptoti c $\\sigma$-moment generating function. Comparing this with our descripti on of the asymptotic distribution of the eigenvalues of a Haar-random orth ogonal matrix\, we obtain an equidistribution result that is "sideways" co mpared to the equidistribution results obtained by Katz and Sarnak\, i.e. where the order of the limits in $d$ and $q$ have been exchanged. This sid eways equidistribution is finer in that it sees the stable cohomology of l ocal systems in all degrees instead of just the zeroth degree needed to co mpute monodromy. \n\nThe techniques used are robust and apply also to the L-functions of more general smooth hypersurface sections\, as well as some simple Dirichlet characters that were previously studied by Bergström-Di aconu-Petersen-Westerland. Time permitting\, we will briefly discuss furth er generalizations and related work in progress joint with Bertucci / Bilu / Bilu and Das.\n LOCATION:https://researchseminars.org/talk/MITNT/108/ END:VEVENT END:VCALENDAR