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Алгебра и анализ, 2017, том 29, выпуск 1, страницы 110–144
(Mi aa1524)
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Эта публикация цитируется в 3 научных статьях (всего в 3 статьях)
Статьи
Endomorphism rings of reductions of elliptic curves and Abelian varieties
Yu. G. Zarhin Department of Mathematics, Pennsylvania State University, University Park, PA 16802, USA
Аннотация:
Let $E$ be an elliptic curve without CM that is defined over a number field $K$. For all but finitely many non-Archimedean places $v$ of $K$ there is a reduction $E(v)$ of $E$ at $v$ that is an elliptic curve over the residue field $k(v)$ at $v$. The set of $v$'s with ordinary $E(v)$ has density 1 (Serre). For such $v$ the endomorphism ring $\operatorname{End}(E(v))$ of $E(v)$ is an order in an imaginary quadratic field.
We prove that for any pair of relatively prime positive integers $N$ and $M$ there are infinitely many non-Archimedean places $v$ of $K$ such that the discriminant $\boldsymbol\Delta(\mathbf v)$ of $\operatorname{End}(E(v))$ is divisible by $N$ and the ratio $\frac{\boldsymbol\Delta(\mathbf v)}N$ is relatively prime to $NM$. We also discuss similar questions for reductions of Abelian varieties.
The subject of this paper was inspired by an exercise in Serre's "Abelian $\ell$-adic representations and elliptic curves" and questions of Mihran Papikian and Alina Cojocaru.
Ключевые слова:
absolute Galois group, Abelian variety, general linear group, Tate module, Frobenius element.
Поступила в редакцию: 10.02.2016
Образец цитирования:
Yu. G. Zarhin, “Endomorphism rings of reductions of elliptic curves and Abelian varieties”, Алгебра и анализ, 29:1 (2017), 110–144; St. Petersburg Math. J., 29:1 (2018), 81–106
Образцы ссылок на эту страницу:
https://www.mathnet.ru/rus/aa1524 https://www.mathnet.ru/rus/aa/v29/i1/p110
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Страница аннотации: | 302 | PDF полного текста: | 81 | Список литературы: | 35 | Первая страница: | 9 |
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