Quadratic forms and elliptic curves. IV
The author obtains some new aspects in his story on ``Quadratic forms and elliptic curves'' [Part I: ibid. 72, No. 7, 157-159 (1996; Zbl 0877.14020); II: ibid. 72, No. 8, 194-196 (1996; Zbl 0877.14021); III: ibid. 72, No. 9, 204-205 (1996; Zbl 0877.14022)]. As in ``I'' he offers by the ``Hopf construction'' a new family of elliptic curves over \(\mathbb{Q}\) with points of infinite order. As special examples he considers \(F_n\), the set of primes \(p\), \((p,2n)=1\), and \(p= a^2+ nb^2\), and the subset \(G_n\) with \(p= 4a^2+1\). He conjectures, supported by machine computation, that \(G_n\) is finite for all \(n\). The point \(P_0= (1,p)\) lies on the elliptic curve \[ E_c: \quad Y^2= X^3- (c+1)X^2+ (p^2+c)X. \] One of his theorems: If \(p\in F_n\), and \(P_0\) is a torsion point on \(E_c\), with \(c=1+ 2(a^2- nb^2)\), then \(p\) belongs to \(G_n\). For \(n=1\), \(G_n= \{5\}\) (Fermat), so for all \(p\equiv 1\pmod 4\) \((p=5\) is known), \(P_0\) is of infinite order over \(E_{-p}\).
scientific article; zbMATH DE number 1143819