Crystalline local systems

Let K be a p-adic field with absolute Galois group G_K. In 1980’s, Fontaine discovered the notion of being crystalline for finite Q_p-representations of G_K. This captures the property of having good reduction, analogously to being unramified in the l-adic case. The notion of crystalline representation is rational in nature, so it has little meaning for torsion representations. On the other hand, based on their discovery of prismatic site, Bhatt and Scholze recently proved that the category of Z_p-lattices of crystalline representations of G_K is equivalent to the category of prismatic F-crystals on O_K. This is expected to be the first step towards understanding torsion crystalline representations.

In this talk, we will report on a joint work with Du, Liu, Shimizu on generalizing Bhatt-Scholze’s result for families of Galois representations. Such a family naturally arises in studying the arithmetic analogue of variation of Hodge structure. We will explain that for a smooth p-adic formal scheme X over O_K, the category of crystalline Z_p-local systems on the generic fiber of X is equivalent to the category of completed prismatic F-crystals on X.