Simultaneous supersingular reductions of CM elliptic curves
-
Menny Aka
,
Manuel Luethi
and
Andreas Wieser
Abstract
We study the simultaneous reductions at several supersingular primes of elliptic curves with complex multiplication. We show – under additional congruence assumptions on the CM order – that the reductions are surjective (and even become equidistributed) on the product of supersingular loci when the discriminant of the order becomes large. This variant of the equidistribution theorems of Duke and Cornut–Vatsal is an(other) application of the recent work of Einsiedler and Lindenstrauss on the classification of joinings of higher-rank diagonalizable actions.
Funding statement: Manuel Luethi acknowledges the support of the SNF (grants 200021_178958, P1EZP2_181725 and 200021_197045) and of the ISF (grant 1483/16). Philippe Michel is partially supported by a DFG-SNF lead agency program grant 200021L_175755 and by the SNF grant 200021_197045. Andreas Wieser was partially supported by SNF grant 20021_178958, SNF Doc. Mobility grant 195737, and ERC 2020 grant no. 833423.
Acknowledgements
We want to express our gratitude towards the referee, whose careful reading of the manuscript improved the exposition and helped us remove an inaccuracy present in a preliminary version of this article. We also thank Valentin Blomer, Farrell Brumley, Brian Conrad, Henri Darmon, Dick Gross, Jennifer-Jayne Jakob, Ilya Khayutin, Bjorn Poonen, Dinakar Ramakrishnan and Tomer Schlank for helpful discussions. Part of this work was carried out while Philippe Michel was visiting the Department of Mathematics at Caltech, while Manuel Luethi was visiting the Einstein institute of Mathematics at the Hebrew University of Jerusalem, and while the authors were visiting the Hausdorff Research Institute during the program “Dynamics: Topology and Numbers”.
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© 2022 Walter de Gruyter GmbH, Berlin/Boston
Articles in the same Issue
- Frontmatter
- Simultaneous supersingular reductions of CM elliptic curves
- Almost all entries in the character table of the symmetric group are multiples of any given prime
- K-stability of cubic fourfolds
- Nguyen’s tridents and the classification of semigraphical translators for mean curvature flow
- A description of monodromic mixed Hodge modules
- The Lawson surfaces are determined by their symmetries and topology
- CMC hypersurfaces with bounded Morse index
- On Montgomery’s pair correlation conjecture: A tale of three integrals
- Kähler spaces with zero first Chern class: Bochner principle, Albanese map and fundamental groups
Articles in the same Issue
- Frontmatter
- Simultaneous supersingular reductions of CM elliptic curves
- Almost all entries in the character table of the symmetric group are multiples of any given prime
- K-stability of cubic fourfolds
- Nguyen’s tridents and the classification of semigraphical translators for mean curvature flow
- A description of monodromic mixed Hodge modules
- The Lawson surfaces are determined by their symmetries and topology
- CMC hypersurfaces with bounded Morse index
- On Montgomery’s pair correlation conjecture: A tale of three integrals
- Kähler spaces with zero first Chern class: Bochner principle, Albanese map and fundamental groups
