Dan Petersen: Moments of families of quadratic L-functions over function fields [...] (NTWS 206)
Channel: Number Theory Web Seminar
Date: 09/04/2024

Title: Moments of families of quadratic L-functions over function fields via homotopy theory

Abstract: This is a report of joint work with Bergström-Diaconu-Westerland and Miller-Patzt-Randal-Williams. There is a "recipe" due to Conrey-Farmer-Keating-Rubinstein-Snaith which allows for precise predictions for the asymptotics of moments of many different families of L-functions. Our work concerns the CFKRS predictions in the case of the quadratic family over function fields, i.e. the family of all L-functions attached to hyperelliptic curves over some fixed finite field. One can relate this problem to understanding the homology of the braid group with certain symplectic coefficients. With Bergström-Diaconu-Westerland we compute the stable homology groups of the braid groups with these coefficients, together with their structure as Galois representations. We moreover show that the answer matches the number-theoretic predictions. With Miller-Patzt-Randal-Williams we prove a uniform range for homological stability with these coefficients. Together, these results imply the CFKRS predictions for all moments in the function field case, for all sufficiently large (but fixed) q.

Link to slides: https://drive.google.com/file/d/1eac_d7GT998hYlUue_vZKbTsVmBE_Hyl

Number Theory Web Seminar: https://www.ntwebseminar.org

Original air date: 
Thursday, April 4, 2024 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)Friday, April 5, 2024 (2am AEDT, 5am NZDT)