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Finiteness of reductions of Hecke orbits

Authors :
Kisin, Mark
Lam, Yeuk Hay Joshua
Shankar, Ananth N.
Srinivasan, Padmavathi
Publication Year :
2021

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.<br />Comment: 15 pages

Details

Database :
arXiv
Publication Type :
Report
Accession number :
edsarx.2109.05147
Document Type :
Working Paper