# Pointwise good reduction criteria for local systems

# Pointwise good reduction criteria for local systems

Let S be a connected smooth rigid analytic variety over a p-adic field K and let T be a p-adic local system over S. A celebrated theorem of Liu and Zhu says that if V is de Rham at one classical point, then V is globally de Rham. When S has good reduction over O_K, one naturally asks about analogous statements when we replace "de Rham" by "crystalline" or "semistable". It is well known that the naive analogues are false.

In an ongoing joint work with Haoyang Guo (with many contributions by Sasha Petrov), we prove that Liu-Zhu's result can be remedied if one tests at "sufficiently many" points, when we choose a good integral model for S. In particular, if V is crystalline or semistable at every classical point, then it is crystalline or semistable relative to the chosen integral model. As a direct consequence, the notion of crystallinity or semistability of V depends only on the rigid analytic variety S and not on the choice of a good model (provided that there exists one). I will also discuss the l-adic analogue of this result as well as its relation to purity statements. If time permits, I will discuss some speculations yet to be affirmed.