Genus 2 Curves with Quaternionic Multiplication

Abstract

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.

On Local Constants Associated to Arithmetical Automorphic Function

“…(Sketch) Expression (i) follows from the work of Hashimoto-Murabayashi [11], Baba-Granath [2] and Rotger [15], [16]. In fact, Hashimoto-Murabayashi gave a parametric family of curves of genus 2 whose jacobian has quaternionic multiplication by O 6 .…”

“…By comparing the triangle [s(0), s(1), s(∞)] with the triangles we have used to define the functions t + 6 , t (2) 6 , and t (3) 6 , we obtain the corresponding adapted local constants k P at the elliptic points and SCM-points P . They are listed in table 6.…”

“…That is to say, for every γ ∈ Γ, z ∈ H the following equality holds: (2) 6 , j (3) 6 , j (6) 6 : H → P 1 (C) given by…”

“…We use some of them to cut fundamental half-domains, that is, half-domains containing all the vertices of a fundamental domain exactly once. Table 2 lists axes of symmetries and polygons defining fundamental half-domains for the curves X 6 , X (2) 6 , X Table 3. Angles at the vertices of fundamental half-domains (1) 4t…”

Differential equations and expansions for quaternionic modular forms in the discriminant 6 case

“…In [4], these results are extended to give some terms of the Taylor expansions of the j-function at the elliptic points. In [3], modular forms were computed by restricting Hilbert modular forms to a particular Hirzebruch-Zagier cycle. During the preparation of this manuscript, two articles appeared which also address the issue of expansions of quaternionic modular forms, but with notably different methods.…”