Frobenius and the Hodge numbers of the generic fiber

Abstract: For a smooth proper (formal) scheme $\mathfrak{X}$ defined over a valuation ring of mixed characteristic, the crystalline cohomology H of its special fiber has the structure of an F-crystal, to which one can attach a Newton polygon and a Hodge polygon that describe the ''slopes of the Frobenius action on H''. The shape of these polygons are constrained by the geometry of $\mathfrak{X}$ -- in particular by the Hodge numbers of both the special fiber and the generic fiber of $\mathfrak{X}$. One instance of such constraints is given by a beautiful conjecture of Katz (now a theorem of Mazur, Ogus, Nygaard etc.), another constraint comes from the notion of "weakly admissible" Galois representations.

In this talk, I will discuss some results regarding the shape of the Frobenius action on the F-crystal H and the Hodge numbers of the generic fiber of $\mathfrak{X}$, along with generalizations in several directions. In particular, we give a new proof of the fact that the Newton polygon of the special fiber of $\mathfrak{X}$ lies on or above the Hodge polygon of its generic fiber, without appealing to Galois representations. A new ingredient that appears is (a generalized version of) the Nygaard filtration of the prismatic/Ainf cohomology, developed by Bhatt, Morrow and Scholze.

number theory

Audience: researchers in the topic

Harvard number theory seminar