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## 2018 - French computer algebra days - Journées nationales de calcul formel

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## Endomorphism algebras of geometrically split abelian surfaces over Q

An abelian surface defined over $\mathbb{Q}$ is said to be geometrically split if its base change to the complex numbers is isogenous to a product of elliptic curves. In this talk we will determine the algebras that arise as geometric endomorphism algebras of geometrically split abelian surfaces defined over $\mathbb{Q}$. In particular, we will show that there are 92 of them. A key step is determining the set of imaginary quadratic fields $M$ for which there exists an abelian surface over $\mathbb{Q}$ which is geometrically isogenous to the square of an elliptic curve with CM by $M$.

This is joint work with Francesc Fité.

### Citation data

• DOI 10.24350/CIRM.V.20006903
• Cite this video Guitart Xavier (2/28/23). Endomorphism algebras of geometrically split abelian surfaces over Q. CIRM. Audiovisual resource. DOI: 10.24350/CIRM.V.20006903
• URL https://dx.doi.org/10.24350/CIRM.V.20006903

### Bibliography

• FITÉ, Francesc et GUITART, Xavier. Endomorphism algebras of geometrically split abelian surfaces over ℚ. Algebra & Number Theory, 2020, vol. 14, no 6, p. 1399-1421. - https://doi.org/10.48550/arXiv.1807.10010

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