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On \(\theta\)-congruent numbers over real number fields. (English) Zbl 1468.11124

Let \(0<\theta<\pi\) be an angle with rational cosine \(\cos\theta=s/r\), where \(0< |s| < r\) and \(\gcd(r, s) = 1\). Let \((u, v, w)_{\theta}\) denote a triangle with an angle \(\theta\) between the sides \(u\) and \(v\). A positive integer \(n\) is called a \(\theta\)-congruent number if there exists a triangle \((u, v, w)_{\theta}\) satisfying \[ 2rn=uv,\quad w^2=u^2+v^2-2uv\frac{s}{r}. \] The \(\theta\)-congruent numbers with \(\theta = \pi/2\) are just the classical congruent numbers. One can prove that an integer \(n\) is \(\theta\)-congruent if and only if an associated elliptic curve \[ E_{n,\theta}:\;y^2=(x+(r+s)n)(x-(r-s)n), \] has a point of order grater then 2.
In completely analogous manner one can define the notion of \((K,\theta)\)-congruent number, where \(K\) is a given algebraic number field, i.e., we ask about the existence of solutions of the equation defining \(E_{n,\theta}\) in \(K\). It is clear that the existence of point of order \( > 2\) can be too weak condition for \(n\) to be \((K, \theta)\)-congruent. In this paper, the authors study the problem of existence of \((K,\theta)\)-congruent numbers over multi-quadratic number fields, over real number fields of degree co-prime to 6 and over real cubic fields. In each case, under small assumptions on \(n\), it is proved that the number \(n\) is \((K,\theta)\)-congruent if and only if the curve \(E_{n,\theta}\) has positive \(K\)-rank, i.e., the set of \(K\)-rational points on \(E_{n,\theta}\) is infinite.

MSC:

11G05 Elliptic curves over global fields
11R21 Other number fields
11R16 Cubic and quartic extensions
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