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Endomorphism algebras of geometrically split abelian surfaces over Q

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Authors : Guitart, Xavier (Author of the conference)
CIRM (Publisher )

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Abstract : 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.

Keywords : endomorphism algebras; abelian varieties; Q-curves

MSC Codes :
11G10 - Abelian varieties of dimension >1
14K15 - Arithmetic ground fields
14K22 - Complex multiplication, See also {11G15}

Additional resources :
https://www.cirm-math.fr/RepOrga/2805/Slides/guitart_count.pdf

    Information on the Video

    Film maker : Petit, Jean
    Language : English
    Available date : 13/03/2023
    Conference Date : 28/02/2023
    Subseries : Research talks
    arXiv category : Number Theory
    Mathematical Area(s) : Number Theory
    Format : MP4 (.mp4) - HD
    Video Time : 00:46:46
    Targeted Audience : Researchers ; Graduate Students ; Doctoral Students, Post-Doctoral Students
    Download : https://videos.cirm-math.fr/2023-02-28_Guitart.mp4

Information on the Event

Event Title : COUNT - Computations and their Uses in Number Theory / Les calculs et leurs utilisations en théorie des nombres
Event Organizers : Anni, Samuele ; Allombert, Bill ; Balakrishnan, Jennifer ; Bruin, Peter ; Kilicer, Pinar ; Streng, Marco
Dates : 27/02/2023 - 03/03/2023
Event Year : 2023
Event URL : https://conferences.cirm-math.fr/2805.html

Citation Data

DOI : 10.24350/CIRM.V.20006903
Cite this video as: Guitart, Xavier (2023). Endomorphism algebras of geometrically split abelian surfaces over Q. CIRM. Audiovisual resource. doi:10.24350/CIRM.V.20006903
URI : http://dx.doi.org/10.24350/CIRM.V.20006903

See Also

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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