Degrees of parametrizations of elliptic curves by Shimura curves

Authors
Citation
S. Takahashi, Degrees of parametrizations of elliptic curves by Shimura curves, J NUMBER TH, 90(1), 2001, pp. 74-88
Citations number
20
Language
INGLESE
art.tipo
Article
Categorie Soggetti
Mathematics
Journal title
JOURNAL OF NUMBER THEORY
ISSN journal
0022-314X → ACNP
Volume
90
Issue
1
Year of publication
2001
Pages
74 - 88
Database
ISI
SICI code
0022-314X(200109)90:1<74:DOPOEC>2.0.ZU;2-V
Abstract
In this paper. we study some properties of parametrizations of elliptic cur ves 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 ord er 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 con nected kernel. For a prime r \N, we study the map on groups of connected co mponents of Neron fibers at r induced from J --> E. We show that if r divid es D. then the map is surjective. Moreover, we study some relations among d egrees of parametrizations X-0(D)(M) --> E when D and M vary. Also. we desc ribe a method of computing the degree of X-0(D)(M) --> E when D > 1. (C) 20 01 Academic Press.