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Degrees of Parametrizations of Elliptic Curves by Shimura Curves

✍ Scribed by Shuzo Takahashi

Publisher: Elsevier Science
Year: 2001
Tongue: English
Weight: 146 KB
Volume: 90
Category: Article
ISSN: 0022-314X
DOI: 10.1006/jnth.2000.2614

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✦ Synopsis

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 X D 0 (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 Ä E having the connected kernel. For a prime r | N, we study the map on groups of connected components of Ne ron fibers at r induced from J Ä E. We show that if r divides D, then the map is surjective. Moreover, we study some relations among degrees of parametrizations X D 0 (M) Ä E when D and M vary. Also, we describe a method of computing the degree of X D 0 (M ) Ä E when D>1.

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