Concordant pairs in ratios with rank at least two and the distribution of \(\theta\)-congruent numbers. (English) Zbl 1496.11079

Let \(M\) and \(N\) are distinct non-zero integers such that their GCD is square-free. Let \(E(M,N)\) be the elliptic curve defined by the equation: \(y^2=x(x+M)(x+N)\). The pair \((M,N)\) is called a concordant pair if the concordant forms \(X^2+MY^2=Z^2, X^2+NY^2=W^2\) have a common integral solution \((X,Y,Z,W)\) such that \(\mathrm{gcd}(X,Y)=1\) and \(XYZ\ne 0\). It is well known that \((M,N)\) is a concordant pair if and only if \(E(M,N)\) has a nontrivial rational point of order different from \(2\). Let \(k\) and \(\ell\) be distinct nonzero integers and \(m\) a positive integer. The author shows that for any integer \(a\) there are infinitely many integers \(n\equiv a \mod m\), inequivalent modulo squares, such that the rank of \(E(M,N) (M=kn,N=\ell n)\) is at least \(2\). Taking \(a=1\), there are infinitely many concordant pairs \((M,N)\) with \(M\equiv k \mod m\) and \(N\equiv \ell \mod m\). Further he shows that there exist positive real numbers \(C_1\) and \(C_2\) depending on \(k\) and \(\ell\) such that if \(T>C_1\), then the number of square-free integers \(n\) with \(|n|\le T\) for which \(E(kn,\ell n)\) has rank at least \(2\) is at least \(C_2T^{2/7}\). In parallel, he shows comparable results for \(\theta\)-congruent numbers \(n\) by letting \(\cos \theta=s/r\) and \(M=(s+r)n,N=(s-r)n\). Here an integer \(n\) is said to be a \(\theta\)-congruent number if there is a triangle with rational sides of angle \(\theta\) such that \(\cos \theta=s/r\) and area \(n\sqrt{r^2-s^2}\). When \(\theta=\pi/2~(s=0)\), it is a congruent number. To show the results he gives a quadratic twist by a polynomial \(d(t)\in\mathbb Z[k,\ell][t]\) of an elliptic curve, \(\mathbb Q\)-isomorphic to \(E(kn,\ell n)\), which has two \(\mathbb Q(t)\)-rational independent points and uses the injective property of specialization map.


11G05 Elliptic curves over global fields
11D09 Quadratic and bilinear Diophantine equations
11D45 Counting solutions of Diophantine equations
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