Carlo Pagano: On Chowla's non-vanishing conjecture over function fields (NTWS 175)
Channel: Number Theory Web Seminar
Date: 2023-06-12

Abstract: A conjecture of Chowla postulates that no L-function of Dirichlet characters over the rationals vanishes at s=1/2. Soundararajan has proved non-vanishing for a positive proportion of quadratic characters. Over function fields Li has discovered that Chowla's conjecture fails for infinitely many distinct quadratic characters. However, on the basis of the Katz--Sarnak heuristics, it is still widely believed that one should have non-vanishing for 100% of the characters in natural families (such as the family of quadratic characters). Works of Bui--Florea, David--Florea--Lalin, Ellenberg--Li--Shusterman, among others, provided evidence giving a positive proportion of non-vanishing in several such families. I will present an upcoming joint work with Peter Koymans and Mark Shusterman, where we prove that for each fixed q congruent to 3 modulo 4 one has 100% non-vanishing in the family of imaginary quadratic function fields. 


Link to slides: https://drive.google.com/file/d/1qCc5U1Lk8KXgcK7vpjS36NQwrvob2NjO

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

Original air date: 
Thursday, June 8, 2023 (8am PDT, 11am EDT, 4pm GMT, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, June 9, 2023 (1am AEST, 3am NZST)