## 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.

### MSC:

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

### Keywords:

concordant forms; elliptic curve; rank
Full Text:

### References:

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