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# Genus 2 Curves with Quaternionic Multiplication

Published:2008-08-01
Printed: Aug 2008
• Srinath Baba
• H\aa kan Granath
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## Abstract

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 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: Shimura curve, canonical model, quaternionic multiplication, modular form, field of moduli
 MSC Classifications: 11G18 - Arithmetic aspects of modular and Shimura varieties [See also 14G35] 14G35 - Modular and Shimura varieties [See also 11F41, 11F46, 11G18]

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