Proceedings of the Japan Academy, Series A, Mathematical Sciences

On the distribution of rank two $\tau$-congruent numbers

Chad Tyler Davis

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Abstract

A positive integer $n$ is the area of a Heron triangle if and only if there is a non-zero rational number $\tau$ such that the elliptic curve \begin{equation*} E_{τ}^{(n)}: Y^{2} = X(X-nτ)(X+nτ^{-1}) \end{equation*} has a rational point of order different than two. Such integers $n$ are called $\tau$-congruent numbers. In this paper, we show that for a given positive integer $p$, and a given non-zero rational number $\tau$, there exist infinitely many $\tau$-congruent numbers in every residue class modulo $p$ whose corresponding elliptic curves have rank at least two.

Article information

Source
Proc. Japan Acad. Ser. A Math. Sci., Volume 93, Number 5 (2017), 37-40.

Dates
First available in Project Euclid: 29 April 2017

Permanent link to this document
https://projecteuclid.org/euclid.pja/1493431381

Digital Object Identifier
doi:10.3792/pjaa.93.37

Mathematical Reviews number (MathSciNet)
MR3645658

Zentralblatt MATH identifier
1374.14028

Subjects
Primary: 14H52: Elliptic curves [See also 11G05, 11G07, 14Kxx]
Secondary: 11G05: Elliptic curves over global fields [See also 14H52]

Keywords
Elliptic curve $\tau$-congruent number rank

Citation

Davis, Chad Tyler. On the distribution of rank two $\tau$-congruent numbers. Proc. Japan Acad. Ser. A Math. Sci. 93 (2017), no. 5, 37--40. doi:10.3792/pjaa.93.37. https://projecteuclid.org/euclid.pja/1493431381


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