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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

##### MSC:
 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