Rational points on \(X^ +_ 0(N)\) and quadratic \(\mathbb Q\)-curves.
Let \(N\) be an integer \(>1\) and \(X_0(N)\) be the modular curve whose non-cusp points classify the isomorphism classes of isogenies between elliptic curves \(\varphi:E\to E'\) of degree \(N\) with cyclic kernel. We denote by \(X^+_0(N)\) the quotient of \(X_0(N)\) by the Fricke involution \(W_N\). The curve \(X^+_0(N)\) has a model defined over \(\mathbb{Q}\). Rational points of \(X^+_0(N)\) which do not arise from elliptic curves with complex multiplication are called ``exceptional''. \textit{F. Momose} [J. Fac. Sci., Univ. Tokyo, Sect. 1 A Math. 33, 441--466 (1986; Zbl 0621.14018), J. Math. Soc. Japan 39, 269--286 (1987; Zbl 0623.14009)] has given interesting results on the \(\mathbb{Q}\)-rational points of \(X^+_0(N)\). The author of the paper under review gave a computational study of the case when \(N\) is a prime [\textit{S. D. Galbraith}, Exp. Math. 8, 311--318 (1999; Zbl 0960.14010)]. In this paper he continues his previous work studying some of the cases not covered by the results of Momose. Exceptional rational points are found in the cases \(N=91\), \(N=125\) and the \(j\)-invariants of the corresponding quadratic \(\mathbb{Q}\)-curves are obtained.
scientific article; zbMATH DE number 1943335
scientific article