Title: p-Adic Periods for Mixed Motives

Abstract: A 'period' is the integral of an algebraic differential form along a rational homology cycle (possibly relative to an algebraic subvariety). The simplest example is 2πi, the integral of dz/z around the unit circle in the complex affine line, which is a period (in the usual sense) of the complex exponential. Algebraic change of variables, and more generally morphisms between algebraic varieties, give relations between periods. Grothendieck and later Kontsevich--Zagier made period conjectures, saying roughly that all relations between periods should come from maps of algebraic varieties. Such conjectures imply, for example, that all values of the Riemann zeta function at positive odd integers should be transcendental and even algebraically independent from one another.


Periods have an obvious connection to cohomology, and they can ultimately be expressed in terms of the torsor of paths between the de Rham and Betti fiber functors on categories of (mixed) motives. There is a special complex "period point" of this torsor corresponding to integration of differential forms along homology cycles, and the period conjectures say that this point is dense in the torsor.


There is a different notion of integration, with p-adic rather than real or complex values, due to Coleman. Deligne and Chatzistamatiou--Unver constructed p-adic period points that lead to Coleman integrals in the special case of mixed Tate motives, and Yamashita made an analogous p-adic period conjecture. We review all of this and then discuss recent work of myself and Ishai Dan-Cohen developing p-adic period points and conjectures for general mixed motivic structures, with a view toward applications to the Chabauty--Kim method and syntomic regulators.


Panopto link: https://huji.cloud.panopto.eu/Panopto/Pages/Viewer.aspx?id=f67a7abb-3a78-4bfd-a0e7-b3a50068a360