CM-lifts for Isogeny Classes of Shimura F-crystals over Finite Fields

TL;DR

This paper extends results on lifting isogeny classes of abelian varieties with complex multiplication from finite fields to larger contexts involving Shimura varieties of Hodge type, connecting characteristic p structures with characteristic 0 lifts.

Contribution

It generalizes Zink's result to broader settings involving Shimura varieties of Hodge type, establishing new lift existence results for isogeny classes with complex multiplication.

Findings

Existence of lifts to characteristic 0 for certain isogeny classes over finite fields.

Extension of Zink's results to Shimura varieties of Hodge type.

Connections between Frobenius endomorphisms and complex multiplication in mixed characteristic.

Abstract

We extend to large contexts pertaining to Shimura varieties of Hodge type a result of Zink on the existence of lifts to characteristic 0 of suitable representatives of certain isogeny classes of abelian varieties endowed with Frobenius and other endomorphisms over $\dbF_{p^q}$, whose $p$-divisible groups in mixed characteristic $(0,p)$ are with complex multiplication.