On a conjecture of Gross and Zagier

TL;DR

This paper proves a conjecture by Gross and Zagier that relates the torsion subgroup of rational elliptic curves to Tamagawa numbers, Manin constants, and Tate-Shafarevich groups over imaginary quadratic fields.

Contribution

It provides a proof of the conjecture, establishing a key relationship in the arithmetic of elliptic curves over imaginary quadratic fields.

Findings

The conjecture by Gross and Zagier is confirmed to be true.

The order of the torsion subgroup divides the specified product for rank 1 elliptic curves.

The result advances understanding of the arithmetic invariants of elliptic curves.

Abstract

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.