Finiteness of reductions of Hecke orbits

TL;DR

This paper establishes finiteness results for reductions of Hecke orbits of abelian varieties over local fields, with implications for CM lifts of supersingular varieties and K3 surfaces, using p-adic Hodge theory and group techniques.

Contribution

It proves new finiteness theorems for Hecke orbit reductions and their implications for CM lifts of supersingular abelian varieties and K3 surfaces.

Findings

Finiteness of supersingular abelian varieties admitting CM lifts.

Finiteness of supersingular K3 surfaces with CM lifts.

Application of p-adic Hodge theory and group methods.

Abstract

We prove two finiteness results for reductions of Hecke orbits of abelian varieties over local fields: one in the case of supersingular reduction and one in the case of reductive monodromy. As an application, we show that only finitely many abelian varieties on a fixed isogeny leaf admit CM lifts, which in particular implies that in each fixed dimension $g$ only finitely many supersingular abelian varieties admit CM lifts. Combining this with the Kuga-Satake construction, we also show that only finitely many supersingular $K3$-surfaces admit CM lifts. Our tools include $p$-adic Hodge theory and group theoretic techniques.