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Rational points on the modular curves \(X^ +_ 0(p^ r)\). (English) Zbl 0621.14018
The paper is concerned with determining the \({\mathbb{Q}}\)-rational points on the modular curves \(X^+_ 0(p^ r)\), where \(X^+_ 0(N)\) denotes the quotient of \(X_ 0(N)\) by the Atkin-Lehner involution \(w_ N\). The main result is that when \(p=2, 3, 7, 11\) or \(\geq 17\), \(r\geq 2\), the rank of \(J^-_ 0({\mathbb{Q}})\) is finite and the genus of \(X^+_ 0(p^ r)\) is greater than 0 then there are no non-cuspidal \({\mathbb{Q}}\)-rational points except those coming from elliptic curves with complex multiplication. The basic idea is to consider the inverse image of a possible \({\mathbb{Q}}\)-rational point on \(X_ 0(p^ r)\). This inverse image consists of two points defined over a quadratic field and a close study is made of the corresponding elliptic curves. In particular, a study at primes over p reveals that if the image of a \({\mathbb{Q}}\)-rational point on \(X^+_ 0(p^ r)\) has finite order in \(J^-_ 0({\mathbb{Q}})\) then this image is necessarily 0 and a general argument shows that any such point is a CM-point.
Reviewer: T.Ekedahl

14G05 Rational points
14H25 Arithmetic ground fields for curves
14G25 Global ground fields in algebraic geometry
14K22 Complex multiplication and abelian varieties
11F06 Structure of modular groups and generalizations; arithmetic groups