William Duke, On the analytic theory of isotropic ternary quadratic forms (NTWS 152)
Channel: Number Theory Web Seminar
Date: 2023-05-19

Abstract: I will describe recent work giving an asymptotic formula for a count of primitive integral zeros of an isotropic ternary quadratic form in an orbit under integral automorphs of the form. The constant in the asymptotic is explicitly computed in terms of local data determined by the orbit.  This is compared  with the well-known asymptotic for the  count of all primitive zeros. Together with an extension  of results of Kneser by R. Schulze-Pillot on the classes  in a genus of representations, this yields a formula for the number of  orbits,  summed over a genus of forms,  in terms of the number of local orbits. For a certain special class of forms a simple explicit formula is given for this number.

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

Original air date: 
Thursday, December 15, 2022 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm Indian Standard Time)
Friday, December 16, 2022 (12am CST, 3am AEDT, 5am NZDT)