Rational points on the modular curves \(X^ +_ 0(N)\)
The author studies the non-cuspidal rational points on the curve \(X^+_ 0(N)\), which is the quotient of the modular curve \(X_ 0(N)\) by the action of the Atkin-Lehner involution \(w_ N\). From the modular interpretation of \(X_ 0(N)\) it is easy to see that there is a natural morphism \(f^+: X^+_ 0(N)\to J^-_ 0(p)=J_ 0(p)/(1+w_ p)\cdot J_ 0(p)\) when p is a divisor of N and \(J_ 0(p)\) the Jacobian of \(X_ 0(p)\). Using a result of \textit{B. Mazur} and \textit{M. Rapoport} [``Modular curves and the Eisenstein ideal'', Appendix, Publ. Math., Inst. Hautes Étud. Sci. 47 (1977), 33-186 (1978; Zbl 0394.14008)] the author shows that if p is a prime divisor of N, and \(x\in X^+_ 0(N)\) is a non- cuspidal rational point, then \(f^+(x)| {\mathbb{F}}_ p\) is the unit section. Under the assumption that \(p=11\), or \(p\geq 17\) and \(J^-_ 0(p)({\mathbb{Q}})\) is finite one sees that \(f^+(x)\) is actually the unit section. This follows immediately from the fact that the image of the cuspidal divisor class group in \(J^-_ 0(p)\) is an étale group scheme, which is in fact the entire torsion subgroup of the Mordell-Weil group of \(J^-_ 0(p)\) (loc. cit. p. 143). Thus, \(f^+(x)\) generates an étale subgroup of \(J^-_ 0(p)\), and so \(f^+(x)\) is the unit section precisely when \(f^+(x)| {\mathbb{F}}_ p\) is so as well. When \(p\neq 37\) this implies that an elliptic curve that represents the \(point\quad x\) has complex multiplication. Hence, under the conditions on N and p given above, the only non-cuspidal rational points on \(X^+_ 0(N)\) are the C.M. points. This paper extends earlier work of the author on \(X_{split}(p)\) [Compos. Math. 52, 115-137 (1984; Zbl 0574.14023)] and \(X^+_ 0(p^ r)\) [to appear].
scientific article; zbMATH DE number 4010665
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