On the parity of the index of ramified Heegner divisors

Abstract

Let E be an elliptic curve defined over the rationals and let N be its conductor. Assume N is prime. In this paper, we prove that the index on E of the Heegner divisor of discriminant \(D=-~4N\) is even provided \(N\equiv 7\pmod {8}\) and discuss some conjectures on further parity properties for the indexes on E of Heegner divisors of discriminant D dividing 4N. One of these conjectures suggests a possible link between the parity of the eigenvalue \(a_A(2)\) and the parity of the Safarevic-Tate group Open image in new window of certain elliptic curves A of square conductor.