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Title: Endomorphism algebras of geometrically split abelian surfaces over $Q$
Author: Fité Naya, Francesc
Guitart Morales, Xavier
Keywords: Teoria de nombres
Varietats de Shimura
Varietats abelianes
Geometria algebraica
Number theory
Shimura varieties
Abelian varieties
Algebraic geometry
Issue Date: 30-Jul-2020
Publisher: Mathematical Sciences Publishers
Abstract: We determine the set of geometric endomorphism algebras of geometrically split abelian surfaces defined over $\mathbb{Q}$. In particular we find that this set has cardinality 92 . The essential part of the classification consists in determining the set of quadratic imaginary fields $M$ with class group $\mathrm{C}_2 \times \mathrm{C}_2$ for which there exists an abelian surface $A$ defined over $\mathbb{Q}$ which is geometrically isogenous to the square of an elliptic curve with CM by $M$. We first study the interplay between the field of definition of the geometric endomorphisms of $A$ and the field $M$. This reduces the problem to the situation in which $E$ is a $\mathbb{Q}$ curve in the sense of Gross. We can then conclude our analysis by employing Nakamura's method to compute the endomorphism algebra of the restriction of scalars of a Gross $\mathbb{Q}$-curve.
Note: Reproducció del document publicat a:
It is part of: Algebra & Number Theory, 2020, vol. 14, num. 6, p. 1399-1421
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ISSN: 1937-0652
Appears in Collections:Articles publicats en revistes (Matemàtiques i Informàtica)

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