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Genus 2 Curves with Quaternionic Multiplication

Published online by Cambridge University Press:  20 November 2018

Srinath Baba
Affiliation:
Department of Mathematics and Statistics, Concordia University, Montréal, QC, H3G 1M8 e-mail:sbaba@mathstat.concordia.ca
Håkan Granath
Affiliation:
Department of Mathematics, Karlstad University, 65188 Karlstad, Sweden e-mail:hakan.granath@kau.se
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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.

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 2008

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