Tamagawa number conjecture for a $p$-adic family of uniform $F$-crystals

TL;DR

This paper extends K. Kato's work on syntomic complexes and their relation to $L$-function special values from a single $F$-crystal to a $p$-adic family of such crystals, advancing the understanding of their arithmetic properties.

Contribution

It generalizes Kato's results to a $p$-adic family of $F$-crystals, providing new insights into their cohomology and $L$-function relations.

Findings

Extended syntomic complex constructions to $p$-adic families

Established links between cohomology of families and $L$-values

Generalized Tamagawa number conjecture in this setting

Abstract

K. Kato has recently constructed certain syntomic complexes associated with a uniform $F$-crystal over a smooth project variety $X$ and related their cohomology groups to special values of the $L$-function attached to $F$. In this paper, we generalize his results to a $p$-adic family of $F$-crystals.