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Abstract We complete the computation of all$$\mathbb {Q}$$ -rational points on all the 64 maximal Atkin-Lehner quotients$$X_0(N)^*$$ such that the quotient is hyperelliptic. To achieve this, we use a combination of various methods, namely the classical Chabauty–Coleman, elliptic curve Chabauty, quadratic Chabauty, and the bielliptic quadratic Chabauty method (from a forthcoming preprint of the fourth-named author) combined with the Mordell-Weil sieve. Additionally, for square-free levelsN, we classify all$$\mathbb {Q}$$ -rational points as cusps, CM points (including their CM field andj-invariants) and exceptional ones. We further indicate how to use this to compute the$$\mathbb {Q}$$ -rational points on all of their modular coverings.
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Building on Mazur’s 1978 work on prime degree isogenies, Kenku determined in 1981 all possible cyclic isogenies of elliptic curves over . Although more than 40 years have passed, the determination of cyclic isogenies of elliptic curves over a single other number field has hitherto not been realised. In this paper we develop a procedure to assist in establishing such a determination for a given quadratic field. Executing this procedure on all quadratic fields with we obtain, conditional on the Generalised Riemann Hypothesis, the determination of cyclic isogenies of elliptic curves over quadratic fields, including and . To make this procedure work, we determine all of the finitely many quadratic points on the modular curves and , which may be of independent interest.more » « less
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We describe how the quadratic Chabauty method may be applied to determine the set of rational points on modular curves of genus$$g>1$$whose Jacobians have Mordell–Weil rank$$g$$. This extends our previous work on the split Cartan curve of level 13 and allows us to consider modular curves that may have few known rational points or non-trivial local height contributions at primes of bad reduction. We illustrate our algorithms with a number of examples where we determine the set of rational points on several modular curves of genus 2 and 3: this includes Atkin–Lehner quotients$$X_0^+(N)$$of prime level$$N$$, the curve$$X_{S_4}(13)$$, as well as a few other curves relevant to Mazur's Program B. We also compute the set of rational points on the genus 6 non-split Cartan modular curve$$X_{\scriptstyle \mathrm { ns}} ^+ (17)$$.more » « less
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