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Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)

 

Modular curves of genus 2


Authors: Enrique González-Jiménez and Josep González
Journal: Math. Comp. 72 (2003), 397-418
MSC (2000): Primary 14G35, 14H45; Secondary 11F11, 11G10
Published electronically: June 4, 2002
MathSciNet review: 1933828
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Abstract: We prove that there are exactly $149$ genus two curves $C$ defined over $\mathbb{Q} $ such that there exists a nonconstant morphism $\pi:X_1(N)\rightarrow C$ defined over $\mathbb{Q} $ and the jacobian of $C$ is $\mathbb{Q} $-isogenous to the abelian variety $A_f$ attached by Shimura to a newform $f\in S_2(\Gamma_1(N))$. We determine the corresponding newforms and present equations for all these curves.


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Additional Information

Enrique González-Jiménez
Affiliation: Department de Matemàtiques, Universitat Autònoma de Barcelona, Bellaterra, Barcelona, E-08193, Spain
Email: enrikegj@mat.uab.es

Josep González
Affiliation: Escola Universitària Politècnica de Vilanova i la Geltrú, Av. Victor Balaguer s/n, E-08800 Vilanova i la Geltrú, Spain
Email: josepg@mat.upc.es

DOI: http://dx.doi.org/10.1090/S0025-5718-02-01458-8
PII: S 0025-5718(02)01458-8
Keywords: Hyperelliptic modular curves
Received by editor(s): October 10, 2000
Received by editor(s) in revised form: April 4, 2001
Published electronically: June 4, 2002
Additional Notes: The first author was supported in part by DGI Grant BHA2000-0180
The second author was supported in part by DGI Grant BFM2000-0794-C02-02
Article copyright: © Copyright 2002 American Mathematical Society