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arXiv · 1210.8231

Congruent Numbers and Heegner Points

Ye Tian
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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.

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Ye Tian. 2012-10-31. Congruent Numbers and Heegner Points. https://arxiv.org/abs/1210.8231

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