Heegner point constructions and fundamental units in cubic fields

TL;DR

This paper uses Heegner points to establish the existence of rational points on certain elliptic curves over cubic fields, linking fundamental units, class numbers, and modular functions, and determining their algebraic ranks.

Contribution

It provides a new expression for the fundamental unit of cubic fields in terms of class number and modular functions, extending classical results to cubic fields.

Findings

Existence of nontorsion rational points on elliptic curves under specified conditions.

Elliptic curve with equation y^2 = x^3 + D has rank 1, while y^2 = x^3 - D has rank 0.

New formula for fundamental units in cubic fields involving class number and modular functions.

Abstract

We use Heegner points to prove the existence of nontorsion rational points on the elliptic curve $y^2 = x^3 + D$ for any rational number $D=a/b$ such that $a$ and $b$ are squarefree integers for which $6$, $a$, and $b$ are pairwise relatively prime, $a\equiv b\pmod{4}$, $\lvert a\rvert\lvert b\rvert^{-1}\equiv5$ or $7\pmod{9}$, and $h_K$ is odd, where $K:=\mathbb{Q}(\sqrt[3]{D})$. In particular, we show that under these assumptions, the elliptic curve with equation $y^2 = x^3 + D$ has algebraic rank $1$ and the elliptic curve with equation $y^2 = x^3 - D$ has algebraic rank $0$. This follows from our new expression for the fundamental unit of $\mathscr{O}_K$ in terms of the class number $h_K$ and the norm of a special value of a modular function of level $6$, for any integer $D$ relatively prime to $6$, not congruent to $\pm1\pmod{9}$, for which no exponent in its prime factorization is a multiple of $3$. This expression is an analogue of a theorem of Dirichlet in 1840 relating the fundamental unit of a real quadratic field to its class number and a product of cyclotomic units.