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Rational points on the modular curves \(X_{split}(p)\). (English) Zbl 0574.14023

In a well known paper [Publ. Math., Inst. Haut. Étud. Sci. 47(1977), 33-186 (1978; Zbl 0394.14008)], B. Mazur shows that if p is a prime number \(\neq 2, 3, 5, 7\), or 13 then \(X_{split}(p)\) has only a finite number of \({\mathbb{Q}}\)-rational points, where \(X_{split}(p)\) is the modular curve classifying elliptic curves with a normalizer of split Cartan structure on their p-division points. In this paper the author shows that if p (\(\neq 13\) or 37) is a prime number \(\geq 11\) such that \(J_ 0(p)^-({\mathbb{Q}})\) is finite then the only non-cuspidal rational points on \(X_{split}(p)\) are the C M points. These are the points corresponding to classes (E,(A,B)) where A and B are two independent subgroups of the elliptic curve E of order p.
The proof involves a careful analysis of the geometry of the modular curves \(X_{split}(p)\) and \(X_{sp.Car\tan}(p)\), which follows along the lines of B. M. Mazur’s study of \(X_ 0(p)\) in Invent. Math. 44, 129-162 (1978; Zbl 0386.14009). Assuming the existence of a non- cuspidal \({\mathbb{Q}}\)-rational point on \(X_{split}(p)\) the author uses his analysis of the modular curves to construct a degree zero divisor class on \(J_ 0(p)\) that does not intersect \(J_ 0(p)^+\). Using the fact that \(J_ 0(p)^-({\mathbb{Q}})\) is finite the author shows that the divisor class in question represents 0 in \(J_ 0(p)\). He is then able to conclude either that the rational point is a C M point, or that \(X_ 0(p)\) is hyperelliptic and has an automorphism other than \(w_ p\). Ogg has shown that the latter situation can occur only for \(p=37.\)
Finally, letting \(\tilde J\) denote the Eisenstein quotient of \(J_ 0(p)\) the author shows that \(\dim J_ 0(p)-\dim \tilde J\) is an upper bound (if \(p\geq 17)\) for the number of non-cuspidal rational points on \(X_{split}(p)\) that are not C M points. There is also a discussion of the exceptional cases \(p=13\) and \(p=37\).
Reviewer: S.Kamienny

MSC:

14G05 Rational points
11F11 Holomorphic modular forms of integral weight
14H45 Special algebraic curves and curves of low genus
14K15 Arithmetic ground fields for abelian varieties
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References:

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