Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

According the the introduction to Mazur's Rational Isogenies of Prime Degree the following question was open in 1978:

Let $N$ be one of the integers 39, 65, 91, 125, or 169. Does the modular curve $X_0(N)$ possess noncuspidal rational points?

It seems likely that this should have been resolved in the past 32 years. Does anyone know of a reference for this?

share|improve this question

2 Answers 2

up vote 6 down vote accepted

Let me make David Brown's answer more explicit.

Theorem (Kenku, 1979): $X_0(39)(\mathbb{Q})$ consists entirely of the ($4$) cuspidal points.

Theorem (Kenku, 1980): $X_0(65)(\mathbb{Q})$ and $X_0(91)(\mathbb{Q})$ each consist entirely of the ($4$) cuspidal points.

Theorem (Kenku, 1980, see also Kenku, 1980): $X_0(169)(\mathbb{Q})$ consists entirely of $2$ rational cuspidal points.

Theorem (Kenku, 1981): $X_0(125)(\mathbb{Q})$ consists entirely of $2$ rational cuspidal points.

share|improve this answer

These are all due to Kenku; see mathscinet.

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.