The discriminant 10 Shimura curve and its associated Heun functions

Abstract

The Shimura curve of discriminant 10 is uniformized by a subgroup of an arithmetic (2,2,2,3) quadrilateral group. We derive the differential structure of the ring of modular forms for the Shimura curve and relate the ring generators to explicit Heun functions for the quadrilateral group. We also show that the Picard–Fuchs equation of the associated family of abelian surfaces has solutions that are modular forms. These results are used to completely describe the exceptional sets of the Heun functions, and we show how to find examples like Hl272,736;112,712,23,12;−9625=21/252/334/3.

Picard–Fuchs equations for Shimura curves over Q

“…In [1], it is shown that the exceptional set of the hypergeometric function is in a one to one correspondence with the isogeny class of a certain elliptic point. Similarly it is shown in [2] that the exceptional set of the Heun function is also represented by an isogeny class. We now show that this phenomenon is more general.…”

“…Thus, if γ t is any loop on E t varying continuously in t, then p(t) = γt ω t is a solution to the Picard-Fuchs equation 4t(t − 1)D 2 p(t) + (8t − 4)Dp(t) + p(t) = 0. This is a hypergeometric differential equation, and its monodromy group is isomorphic to the congruence group Γ (2). Moreover, if we choose t = λ(z), the Legendre modular function, then it can be shown that this differential equation has ϑ 2 3 (z) as a solution, where ϑ 3 (z) is a particular theta function [13].…”

“…The curve is the coarse moduli space of abelian surfaces with quaternionic multiplication (QM) by a maximal order of the indefinite quaternion algebra of discriminant 10 over . In [2], a uniformizer is chosen and the Picard–Fuchs equation is computed. Moreover, it is shown that there is an algebraic modular form of weight 6 such that any branch of is a solution of the Picard–Fuchs equation.…”