Abstract
Let K be an imaginary quadratic field and \(K_\infty \) be the \({\textbf{Z}}_p^2\)-extension of K. Answering a question of Ahmed and Lim, we show that the Pontryagin dual of the Selmer group over \(K_\infty \) associated to a supersingular polarized abelian variety admits an algebraic functional equation. The proof uses the theory of \(\Gamma \)-system developed by Lai, Longhi, Tan and Trihan. We also show the algebraic functional equation holds for Sprung’s chromatic Selmer groups of supersingular elliptic curves along \(K_\infty \).
Résumé
Soit K un corps quadratique imaginaire et \(K_\infty \) l’unique \(\textbf{Z}_p^2\)-extension de K. Répondant à une question d’Ahmed et Lim, nous montrons que le dual de Pontryagin du groupe de Selmer sur \(K_\infty \) associé à une variété abélienne supersingulière et polarisée admet une équation fonctionnelle algébrique. La preuve utilise la théorie des \(\Gamma \)-systèmes développée par Lai, Longhi, Tan et Trihan. Nous démontrons aussi qu’une équation fonctionnelle algébrique tient pour les groupes de Selmer chromatiques de Sprung sur \(K_\infty \) de courbes elliptiques supersingulières.
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Acknowledgements
The author thanks his Ph.D. advisor Antonio Lei for suggesting the problem and for his continuous support while investigating this topic. The author is also grateful to Jeffrey Hatley and Eyal Goren for answering our questions. The author also thanks Meng Fai Lim for his suggestions. The author also extends his thanks to Anthony Doyon for many helpful discussions. Finally, many thanks to Luochen Zhao for pointing out an inaccuracy in the original proof of lemma 3.8 and to the referee for many constructive comments that helped improve the quality of the paper. The author’s research is supported by the Canada Graduate Scholarships - Doctoral program from the Natural Sciences and Engineering Research Council of Canada.
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Dion, C. Functional equations for supersingular abelian varieties over \({\textbf{Z}}_p^2\)-extensions. Ann. Math. Québec 48, 221–251 (2024). https://doi.org/10.1007/s40316-022-00210-z
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DOI: https://doi.org/10.1007/s40316-022-00210-z