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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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This content will become publicly available on August 13, 2026
Kontsevich–Zorich monodromy groups of translation covers of some platonic solids
We compute the Zariski closure of the Kontsevich–Zorich monodromy groups arising from certain square-tiled surfaces that are geometrically motivated. Specifically we consider three surfaces that emerge as translation covers of platonic solids and quotients of infinite polyhedra and show that the Zariski closure of the monodromy group arising from each surface is equal to a power of\text{SL}(2,{\mathbb{R}}). We prove our results by finding generators for the monodromy groups, using a theorem of Matheus–Yoccoz–Zmiaikou (2014) that provides constraints on the Zariski closure of the groups (to obtain an “upper bound”), and analyzing the dimension of the Lie algebra of the Zariski closure of the group (to obtain a “lower bound”). Moreover, combining our analysis with the Eskin–Kontsevich–Zorich formula (2014), we also compute the Lyapunov spectrum of the Kontsevich–Zorich cocycle for said square-tiled surfaces.
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- Award ID(s):
- 2103136
- PAR ID:
- 10639731
- Publisher / Repository:
- EMS Press
- Date Published:
- Journal Name:
- Groups, Geometry, and Dynamics
- Volume:
- 19
- Issue:
- 3
- ISSN:
- 1661-7207
- Page Range / eLocation ID:
- 1129 to 1163
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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