Degrees of Parametrizations of Elliptic Curves by Shimura Curves

Abstract

In this paper, we study some properties of parametrizations of elliptic curves by Shimura curves. Fix a square-free positive integer N and an isogeny class E of elliptic curves of conductor N defined over Q. Consider a pair (D, M) such that N=DM and the number of prime factors of D is even. Let J be the Jacobian of Shimura curve XD0(M) associated with an Eichler order of level M in an indefinite quaternion albebra of discriminant D defined over Q. There is a unique E in E and a homomorphism J→E having the connected kernel. For a prime r∣N, we study the map on groups of connected components of Néron fibers at r induced from J→E. We show that if r divides D, then the map is surjective. Moreover, we study some relations among degrees of parametrizations XD0(M)→E when D and M vary. Also, we describe a method of computing the degree of XD0(M)→E when D>1.