\(\theta\)-congruent numbers and Heegner points (Q5955994)

Let \(\theta\) be a real number with \(0<\theta<\pi\). A natural number \(n\) is said to be \(\theta\)-congruent if there exists a triangle with rational sides and an angle \(\theta\) whose area is \(n\sqrt {r^2-s^2}\), where \(\cos \theta=s/r\) with \(r,s\in \mathbb{Z}\), \(\text{gcd} (r,s)=1\), and \(r>0\). In this paper the authors prove that a prime number congruent to 23 modulo 24 is \(\pi/ 3\)-congruent. One of the ingredients of their proof is a theorem due to \textit{M. Fujiwara} [Number Theory, K. Győry, A. Pethő, V. T. Sós (eds.), de Gruyter, 235-241 (1998; Zbl 0920.11035)], which says that a natural number \(n\) is \(\theta\)-congruent if and only if the elliptic curve \(E_{n,\theta}\) defined by \(y^2=x(x+(r+s)n) (x-(r-s)n)\) has a rational point of order greater than two. Another ingredient is given by a theorem due to \textit{B. J. Birch} [Symp. Math. 15, 441-445 (1975; Zbl 0317.14015)]: Suppose that an elliptic curve \(E\) has a modular parametrization \(\varphi:X_0 (N)\to E\) such that the Atkin-Lehner involution acts non-trivially on \(E\) through \(\varphi\). Suppose further that \(\varphi(0) \notin 2E(\mathbb{Q})\) and \(p\) is a prime number such that \(-p\) is congruent to a square modulo \(4N\). Then the quadratic twist \(E^{(-p)}\) of \(E\) associated with \(\mathbb{Q}(\sqrt{-p})/ \mathbb{Q}\) has infinitely many rational points. Noting that \(E_{p,\pi/3}\) is the quadratic twist \(E^{(-p)}_{1,2 \pi/3}\), and that there exists a parametrization \(\varphi:X_0 (48)\to E_{1,2\pi/3}\), the authors show that \(\varphi (0) \notin 2E_{1,2\pi/3} (\mathbb{Q})\) through an explicit computation, and thereby finish the proof.