On a conjecture of Gross and Zagier

Dongho Byeon, Taekyung Kim, Donggeon Yhee

Published 2015-01-26Version 1

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 except for two explicit families of curves. We show the validity of the conjecture also for these exceptions, under another conjecture of Stein and Watkins.