BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Evan O'Dorney (Notre Dame) DTSTART:20220205T190000Z DTEND:20220205T200000Z DTSTAMP:20260423T024718Z UID:SCNTD2022/2 DESCRIPTION:Title: Reflection theorems for counting quadratic and cubic polynomials\nby Evan O'Dorney (Notre Dame) as part of Southern California Number Theory D ay\n\nLecture held in APM 6402 and online.\n\nAbstract\nScholz's celebrate d 1932 reflection principle\, relating the 3-torsion in the class groups o f $\\mathbf{Q}(\\sqrt{D})$ and $\\mathbf{Q}(\\sqrt{-3D})$\, can be viewed as an equality among the numbers of cubic fields of different discriminant s. In 1997\, Y. Ohno discovered (quite by accident) a beautiful reflection identity relating the number of binary cubic forms\, equivalently cubic r ings\, of discriminants D and -27D\, where D is not necessarily squarefree . This was proved in 1998 by Nakagawa\, establishing an "extra functional equation" for the Shintani zeta functions counting binary cubic forms. In my talk\, I will present a new and more illuminating method for proving id entities of this type\, based on Poisson summation on adelic cohomology (i n the style of Tate's thesis). Also\, I will present a corresponding refle ction theorem for quadratic polynomials of a quite unexpected shape. The c orresponding Shintani zeta function is in two variables\, counting by both discriminant and leading coefficient\, and finding its analytic propertie s is a work in progress.\n LOCATION:https://researchseminars.org/talk/SCNTD2022/2/ END:VEVENT END:VCALENDAR