Congruent Numbers and Heegner Points

TL;DR

This paper extends the construction of rational points on elliptic curves related to congruent numbers, building on Heegner's work to include integers with multiple prime factors, advancing the understanding of rational solutions in number theory.

Contribution

It generalizes Heegner's method for constructing rational points to a broader class of integers with multiple prime divisors.

Findings

Extended Heegner's results to integers with many prime divisors.

Constructed rational points on elliptic curves for a wider class of integers.

Enhanced understanding of congruent numbers and rational points on elliptic curves.

Abstract

Mohammed Ben Alhocain, in an Arab manuscript of the tenth century, stated that the principal object of the theory of rational right triangles is to find a square which when increased or diminished by a certain number $m$ becomes a square (see Dickson). In modern language, this object is to find a rational point of infinite order on the elliptic curve $my^2=x^3-x$. Heegner constructed (see also Monsky) such rational points in the case that $m$ are primes congruent to 5, 7 modulo 8 or twice primes congruent to 6 modulo 8. We extend Heegner's result to integers $m$ with many prime divisors.