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Finiteness of reductions of Hecke orbits

  • Mark Kisin

    Department of Mathematics, Harvard University, Cambridge, MA 02138, USA

  • Yeuk Hay Joshua Lam

    IHES, 91440 Bures-sur-Yvette, France

  • Ananth N. Shankar

    Department of Mathematics, University of Wisconsin – Madison, Madison, WI 53706, USA

  • Padmavathi Srinivasan

    Department of Mathematics and Statistics, University of Boston, Boston, MA 02215, USA
Finiteness of reductions of Hecke orbits cover

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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 only finitely many supersingular abelian varieties admit CM lifts. Combining this with the Kuga–Satake construction, we also show that only finitely many supersingular K surfaces admit CM lifts. Our tools include -adic Hodge theory and group-theoretic techniques.

Cite this article

Mark Kisin, Yeuk Hay Joshua Lam, Ananth N. Shankar, Padmavathi Srinivasan, Finiteness of reductions of Hecke orbits. J. Eur. Math. Soc. (2023), published online first

DOI 10.4171/JEMS/1395