Heights of Heegner cycles and derivatives of L-series

摘要

In [18], Gross and Zagier proved an identity on modular curves between the height pairings of certain Heegner points and coefficients of certain cusp forms of weight 2. As a consequence, they showed that any modular elliptic curve over an imaginary quadratic field whose L-function has a simple zero as s=1 contains a Heegner point of infinite order. This result plays a crucial rule in the solution of the Gauss class number problem by Goldfeld-Gross=Zagier [15, 18], and in the solution of the Birch and Swinnerton-Dyuer conjecture [24] by Kolyvagin when the L-series of the modular elliptic curve over Q has order≤1.