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Independence of points on elliptic curves arising from special points on modular and Shimura curves, II: local results

Part of: Arithmetic problems. Diophantine geometry Arithmetic algebraic geometry

Published online by Cambridge University Press:  01 May 2009

Alexandru Buium  and
Bjorn Poonen
Alexandru Buium
Affiliation:
Department of Mathematics and Statistics, University of New Mexico, Albuquerque, NM 87131, USA (email: buium@math.unm.edu)
Bjorn Poonen
Affiliation:
Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA (email: poonen@math.mit.edu)
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Abstract

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In the predecessor to this article, we used global equidistribution theorems to prove that given a correspondence between a modular curve and an elliptic curve A, the intersection of any finite-rank subgroup of A with the set of CM-points of A is finite. In this article we apply local methods, involving the theory of arithmetic differential equations, to prove quantitative versions of a similar statement. The new methods apply also to certain infinite-rank subgroups, as well as to the situation where the set of CM-points is replaced by certain isogeny classes of points on the modular curve. Finally, we prove Shimura-curve analogues of these results.

Keywords

modular curveShimura curveCM pointcanonical liftreciprocity law

MSC classification

Secondary: 11G18: Arithmetic aspects of modular and Shimura varieties 14G20: Local ground fields
Type
Research Article
Information
Compositio Mathematica , Volume 145 , Issue 3 , May 2009 , pp. 566 - 602
Copyright
Copyright © Foundation Compositio Mathematica 2009

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