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Quadratic Chabauty for modular curves: algorithms and examples
- Part of
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- Journal:
- Compositio Mathematica / Volume 159 / Issue 6 / June 2023
- Published online by Cambridge University Press:
- 15 May 2023, pp. 1111-1152
- Print publication:
- June 2023
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- Article
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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 
$X_{\scriptstyle \mathrm { ns}} ^+ (17)$.
