Prismatic crystals and crystalline representations in the relative case

Abstract: Let k be a perfect field of characteristic p > 2, and let K be a finite totally ramified extension of W(k)[1/p]. Bhatt-Scholze recently proved that the category of prismatic F-crystals on the absolute prismatic site over O_K is equivalent to the category of lattices of crystalline representations of G_K. We study an analogous situation in the relative case. Let Spf R be an affine p-adic formal scheme smooth over O_K. We show there is a natural faithful functor from the category of certain completed F-crystals on the absolute prismatic site over R to the category of crystalline Z_p-local systems on the generic fiber of Spf R. Furthermore, we show the functor gives an equivalence when R is a formal torus over O_K. This is a joint work with Heng Du, Tong Liu, Koji Shimizu.

number theory

Audience: researchers in the topic

UCLA Number Theory Seminar