Heegner Points on Modular Curves

TL;DR

This paper extends the study of Heegner points to more general modular curves, providing explicit formulas and applications to quadratic twists of $X_0(36)$, with implications for the BSD conjecture.

Contribution

It generalizes Gross's work on Heegner points to broader modular curves and constructs quadratic twists with verified BSD conjecture parts.

Findings

Explicit Gross-Zagier formula for general modular curves

Construction of quadratic twists with Mordell-Weil ranks 0 and 1

Verification of the 2-part of BSD conjecture for these twists

Abstract

In this paper, we study the Heegner points on more general modular curves other than $X_0(N)$, which generalizes Gross' work "Heegner points on $X_0(N)$". The explicit Gross-Zagier formula and the Euler system property are stated in this case. Using such kind of Heegner points, we construct certain families of quadratic twists of $X_0(36)$, with the ranks of Mordell-Weil groups being zero and one respectively, and show that the $2$-part of their BSD conjectures hold.