Title: An arithmetic site of Connes-Consani type for Gaussian integers
Speaker: Aurelien Sagnier (Ecole Polytechnique, France)
Abstract: We are used to seeing integers with the usual structure of an ordered ring. A.Connes and C.Consani proposed in 2014 to look at them with another structure which is an idempotent semiring with an action of the positive integers by multiplication. With the eyes of algebraic geometry, it is a semiringed topos whose points are linked with Riemann zeta function. The hope in the long term is that this new framework coming from algebraic geometry could help translating ideas of the demonstration of the analog of the Riemann hypothesis for zeta functions associated to smooth projective curves over a finite field to the actual Riemann zeta function. I will explain A.Connes' and C.Consani's point of view in the first part of my talk. However, this point of view heavily relies on the natural order on R compatible with addition and multiplication so one may wonder if, for Gaussian integers, where nothing of this sort exists, one can adapt the ideas and the methods of A.Connes and C.Consani. This is what I have done in my Ph.D. thesis and what I will explain in the second part of my talk.
Seminar URL: https://people.math.osu.edu/anderson.2804/gcis/