$$｛\textbf｛Z｝｝_p^2$$Zp2-扩展上的超奇异阿贝尔变种的函数方程

IF 0.5 Q3 MATHEMATICS Annales Mathematiques du Quebec Pub Date : 2023-01-13 DOI:10.1007/s40316-022-00210-z

Cédric Dion

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引用次数: 0

摘要

让 K 是一个虚二次域，$K_\infty $是 K 的 ${\textbf{Z}}_p^2$-扩展。为了回答 Ahmed 和 Lim 提出的一个问题，我们证明了在$K_\infty $上的塞尔默群的庞氏对偶与一个超星极化无边际变种相关联，它承认一个代数函数方程。证明使用了 Lai、Longhi、Tan 和 Trihan 发展的 $\Gamma $-系统理论。我们还证明了沿 $K_infty $的超星椭圆曲线的斯普隆色度塞尔默群的代数函数方程成立。

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Functional equations for supersingular abelian varieties over ${\textbf{Z}}_p^2$-extensions

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 $.

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Annales Mathematiques du Quebec MATHEMATICS-

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1.10

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期刊介绍： The goal of the Annales mathématiques du Québec (formerly: Annales des sciences mathématiques du Québec) is to be a high level journal publishing articles in all areas of pure mathematics, and sometimes in related fields such as applied mathematics, mathematical physics and computer science. Papers written in French or English may be submitted to one of the editors, and each published paper will appear with a short abstract in both languages. History: The journal was founded in 1977 as „Annales des sciences mathématiques du Québec”, in 2013 it became a Springer journal under the name of “Annales mathématiques du Québec”. From 1977 to 2018, the editors-in-chief have respectively been S. Dubuc, R. Cléroux, G. Labelle, I. Assem, C. Levesque, D. Jakobson, O. Cornea. Les Annales mathématiques du Québec (anciennement, les Annales des sciences mathématiques du Québec) se veulent un journal de haut calibre publiant des travaux dans toutes les sphères des mathématiques pures, et parfois dans des domaines connexes tels les mathématiques appliquées, la physique mathématique et l''informatique. On peut soumettre ses articles en français ou en anglais à l''éditeur de son choix, et les articles acceptés seront publiés avec un résumé court dans les deux langues. Histoire: La revue québécoise “Annales des sciences mathématiques du Québec” était fondée en 1977 et est devenue en 2013 une revue de Springer sous le nom Annales mathématiques du Québec. De 1977 à 2018, les éditeurs en chef ont respectivement été S. Dubuc, R. Cléroux, G. Labelle, I. Assem, C. Levesque, D. Jakobson, O. Cornea.

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