In this talk, we study a Tannakian category of admissible pairs, which arise naturally when one is comparing etale and de Rham cohomology of p-adic formal schemes. Admissible pairs are parameterized by local Shimura varieties and their non-minuscule generalizations, which admit period mappings to de Rham affine Grassmannians. After reviewing this theory, we will state a result characterizing the basic admissible pairs that admit CM in terms of transcendence of their periods. This result can be seen as a p-adic analogue of a theorem of Cohen and Shiga-Wolfhart characterizing CM abelian varieties in terms of transcendence of their periods. All work is joint with Sean Howe.

[pre-talk at 1:20PM]