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Mathematics of Computation

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Computing the geometric endomorphism ring of a genus-2 Jacobian


Author: Davide Lombardo
Journal: Math. Comp. 88 (2019), 889-929
MSC (2010): Primary 11F80, 11G10, 11Y99
DOI: https://doi.org/10.1090/mcom/3358
Published electronically: May 11, 2018
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Abstract: We describe an algorithm, based on the properties of the characteristic polynomials of Frobenius, to compute $ \operatorname {End}_{\overline {K}}(A)$ when $ A$ is the Jacobian of a nice genus-2 curve over a number field $ K$. We use this algorithm to confirm that the description of the structure of the geometric endomorphism ring of $ \operatorname {Jac}(C)$ given in the LMFDB ($ L$-functions and modular forms database) is correct for all the genus-2 curves $ C$ currently listed in it. We also discuss the determination of the field of definition of the endomorphisms in some special cases.


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Additional Information

Davide Lombardo
Affiliation: Università di Pisa, Largo Bruno Pontecorvo 5, 56127 Pisa, Italy
Email: davide.lombardo@unipi.it

DOI: https://doi.org/10.1090/mcom/3358
Keywords: Abelian surfaces, Jacobian, Galois representations, endomorphisms
Received by editor(s): January 24, 2017
Received by editor(s) in revised form: December 3, 2017
Published electronically: May 11, 2018
Article copyright: © Copyright 2018 American Mathematical Society

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