Genus 2 Curves with Quaternionic Multiplication

Baba, Srinath ; Granath, Håkan

Canadian journal of mathematics, Tome 60 (2008) no. 4, pp. 734-757

Résumé

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.

Détail

DOI : 10.4153/CJM-2008-033-7

Mots-clés : 11G18, 14G35, Shimura curve, canonical model, quaternionic multiplication, modular form, field of moduli

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