Polynesian Journal of Mathematics

Volume 1, Issue 5

Polarized products of elliptic curves with complex multiplication and field of moduli ℚ

Fabien Narbonne, with an appendix by Francesc Fité and Xavier Guitart

Abstract:

Let $R$ be the maximal order in an imaginary quadratic field $K$. We give an equivalence of categories between the category of polarized abelian varieties isomorphic to a product of elliptic curves over $ℂ$ with complex multiplication (CM) by $R$ and the category of integral hermitian $R$-lattices. Then we apply this equivalence to enumerate all the genus $2$ and $3$ curves with field of moduli $\mathbb{Q}$ and with Jacobian isomorphic to a product of elliptic curves with CM by $R$.

Citation:

Fabien Narbonne, with an appendix by Francesc Fité and Xavier Guitart. Polarized products of elliptic curves with complex multiplication and field of moduli ℚ. Polynesian Journal of Mathematics, Volume 1, Issue 5, Pages 1–32. (Dec. 2024) DOI: 10.69763/polyjmath.1.5

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Milestones:

Received 1 Jul 2024
Revised 17 Dec 2024
Accepted 19 Dec 2024
Published 20 Dec 2024
Communicated by Gaetan Bisson

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