  Gross and Zagier conjectured that if the analytic rank of a rational elliptic
curve is 1, then the order of the rational torsion subgroup of the elliptic
curve divides the product of Tamagawa number, Manin constant, and the square
root of the order of Tate--Shafarevich group over an imaginary quadratic field.
In this paper, we show that this conjecture is true.
