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Endomorphism algebras
of geometrically split abelian surfaces over $\mathbb{Q}$
Francesc Fité and Xavier Guitart |
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Vol. 14 (2020), No. 6, 1399–1421
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Abstract |
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We determine the set of geometric endomorphism algebras of geometrically split abelian surfaces defined over . 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 with class group for which there exists an abelian surface defined over which is geometrically isogenous to the square of an elliptic curve with CM by . We first study the interplay between the field of definition of the geometric endomorphisms of and the field . This reduces the problem to the situation in which is a -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 -curve. |
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Keywords
products of CM elliptic curves, Coleman's conjecture,
endomorphism algebras, singular abelian surfaces
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Mathematical Subject Classification 2010
Primary: 11G18
Secondary: 11G15, 14K22
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Milestones
Received: 8 February 2019
Revised: 31 October 2019
Accepted: 26 February 2020
Published: 30 July 2020
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Authors |
| Francesc Fité | |
| Massachusetts Institute of
Technology Cambridge, MA United States |
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| Xavier Guitart | |
| Universitat de Barcelona Barcelona Catalonia Spain |
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