p-Adic periods and Selmer scheme images

The category of mixed Tate motives over an open integer ring or a number field possesses a notion of p-adic period which diverges somewhat from the complex paradigm: rather than comparing two different fiber functors, it compares two different structures both associated with the same cohomology theory. At first glance, it appears to be a peculiarity of the mixed Tate setting. Yet it plays a central role in the microcosm of mixed Tate Chabauty-Kim. It also connects the study of p-adic iterated integrals with Goncharov’s theory of motivic iterated integrals, and allows us to investigate Goncharov’s conjectures from a p-adic point of view. Lastly, it forms the basis for the so-called p-adic period conjecture. I’ll report on our ongoing work devoted to the construction of p-adic periods beyond the mixed Tate setting, and discuss the possibility of generalizing all aspects of this picture. This is joint work with David Corwin.