Determination of modular elliptic curves by Heegner points

Abstract

For every integer N ≥ 1, consider the set K(N) of imaginary quadratic fields such that, for each K in K(N), its discriminant D is an odd, square-free integer congruent to 1 modulo 4, which is prime to N and a square modulo 4N. For each K, let c = ([x]−[∞]) be the divisor class of a Heegner point x of discriminant D on the modular curve X = X(0)(N) as in [GZ]. (Concretely, such an x is the image of a point z in the upper half plane H such that both z and Nz are roots of integral, definite, binary quadratic forms of the same discriminant D ([B]).) Then c defines a point rational over the Hilbert class field H of K on the Jacobian J = J(0)(N) of X. Denote by cK the trace of c to K.

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