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Genus 2 Curves with Quaternionic Multiplication

Published online by Cambridge University Press:  20 November 2018

Srinath Baba  and
Håkan Granath
Srinath Baba
Affiliation:
Department of Mathematics and Statistics, Concordia University, Montréal, QC, H3G 1M8 e-mail:[email protected]
Håkan Granath
Affiliation:
Department of Mathematics, Karlstad University, 65188 Karlstad, Sweden e-mail:[email protected]
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Abstract

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We explicitly construct the canonical rational models of Shimura curves, both analytically in terms of modular forms and algebraically in terms of coefficients of genus 2 curves, in the cases of quaternion algebras of discriminant 6 and 10. This emulates the classical construction in the elliptic curve case. We also give families of genus 2 $\text{QM}$ curves, whose Jacobians are the corresponding abelian surfaces on the Shimura curve, and with coefficients that are modular forms of weight 12. We apply these results to show that our $j$-functions are supported exactly at those primes where the genus 2 curve does not admit potentially good reduction, and construct fields where this potentially good reduction is attained. Finally, using $j$, we construct the fields of moduli and definition for some moduli problems associated to the Atkin–Lehner group actions.

Keywords

11G1814G35Shimura curvecanonical modelquaternionic multiplicationmodular formfield of moduli
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 60 , Issue 4 , 01 August 2008 , pp. 734 - 757
Copyright
Copyright © Canadian Mathematical Society 2008

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