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Published online by Cambridge University Press: 06 July 2010
Gross to Birch: March 1, 1982
Dear Birch,
I recently found an amusing method to study Heegner points on J0(N). Let E be an elliptic curve over Q of level N, together with a parametrization J0(N) →π E. Let K be a quadratic field of discriminant dK prime to N; let χ be the associated quadratic Dirichlet character and Eχ the twisted curve.
Let F be an imaginary quadratic field in which all prime factors of N split and choose an integral ideal n with and . Assume further that dχ divides dF, so K is contained in H, the Hilbert class field of F. The modular data defines a point of X0(N) rational over H and the divisor gives a point of E(K)−, or equivalently, a rational point on Eχ. One can check that eF is killed by 2 whenever the sign in the functional equation for Eχ is +1. Do your computations support the following?
Conjecture. eF has infinite order iff rank . If this is the case and π is a strong Weil parametrization, let M denote the subgroup generated by the points eF. Then.
I think I can prove that the point eF has infinite order whenever the image of the cuspidal group on E has order divisible by p ≥ 3 and certain p-class groups are trivial. In all these cases, the rank is 1.
Here is a simple case which illustrates the method. Let E = J0(11) and let K be a real quadratic field in which the prime 11 is inert. Choose F as above, and let denote the other imaginary quadratic field contained in FK.
Proposition. If then in and .
Proof. A 5-descent, combined with the fact that hχ ≢ 0 (mod 5), gives an exact sequence
with It will suffice to show as.
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