Finiteness of reductions of Hecke orbits

2021/09/11 by Kisin, Mark, Lam, Yeuk Hay Joshua, Shankar, Ananth N. +1
#14K10 #14K22 #FOS: Mathematics #Number Theory (math.NT) #Primary 11G15

paper · doi:10.48550/arxiv.2109.05147

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.

Related

• Monodromy results for abelian surfaces and K3 surfaces with bad reduction
• Simultaneous supersingular reductions of abelian varieties
• Hecke orbits on Shimura varieties of Hodge type
• Lifts of supersingular abelian varieties with small Mumford-Tate groups
• On counting certain abelian varieties over finite fields