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1 results

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

  • Part of
  • Alexandru Buium, Bjorn Poonen
  • Journal:
    Compositio Mathematica / Volume 145 / Issue 3 / May 2009
    Published online by Cambridge University Press:
    01 May 2009, pp. 566-602
    Print publication:
    May 2009
    • Article
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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.