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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 dimensiongonly finitely many supersingular abelian varieties admit CM lifts. Combining this with the Kuga–Satake construction, we also show that only finitely many supersingular K3surfaces admit CM lifts. Our tools includep-adic Hodge theory and group-theoretic techniques.
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Perfect points on curves of genus 1 and consequences for supersingular K3 surfaces
We describe a method to show that certain elliptic surfaces do not admit purely inseparable multisections (equivalently, that genus 1 curves over function fields admit no points over the perfect closure of the base field) and use it to show that any non-Jacobian elliptic structure on a very general supersingular K3 surface has no purely inseparable multisections. We also describe specific examples of genus 1 fibrations on supersingular K3 surfaces without purely inseparable multisections.
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- PAR ID:
- 10429181
- Date Published:
- Journal Name:
- Compositio Mathematica
- Volume:
- 158
- Issue:
- 5
- ISSN:
- 0010-437X
- Page Range / eLocation ID:
- 1052 to 1083
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
-
Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1112/S0010437X22007382
