We describe how the quadratic Chabauty method may be applied to determine the set of rational points on modular curves of genus
- Award ID(s):
- 1702196
- NSF-PAR ID:
- 10095096
- Date Published:
- Journal Name:
- Compositio Mathematica
- Volume:
- 155
- Issue:
- 6
- ISSN:
- 0010-437X
- Page Range / eLocation ID:
- 1057 to 1075
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
whose Jacobians have Mordell–Weil rank$g>1$ . 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$g$ of prime level$X_0^+(N)$ , the curve$N$ , 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$X_{S_4}(13)$ .$X_{\scriptstyle \mathrm { ns}} ^+ (17)$ -
Abstract Consider a one-parameter family of smooth, irreducible, projective curves of genus $g\ge 2$ defined over a number field. Each fiber contains at most finitely many rational points by the Mordell conjecture, a theorem of Faltings. We show that the number of rational points is bounded only in terms of the family and the Mordell–Weil rank of the fiber’s Jacobian. Our proof uses Vojta’s approach to the Mordell Conjecture furnished with a height inequality due to the 2nd- and 3rd-named authors. In addition we obtain uniform bounds for the number of torsion points in the Jacobian that lie in each fiber of the family.
-
Let K be a number field, and let E/K be an elliptic curve over K. The Mordell–Weil theorem asserts that the K-rational points E(K) of E form a finitely generated abelian group. In this work, we complete the classification of the finite groups which appear as the torsion subgroup of E ( K ) for K a cubic number field. To do so, we determine the cubic points on the modular curves X1(N) for N = 21,22,24,25,26,28,30,32,33,35,36,39,45,65,121. As part of our analysis, we determine the complete lists of N for which J0(N), J1(N), and J1(2,2N) have rank 0. We also provide evidence to a generalized version of a conjecture of Conrad, Edixhoven, and Stein by proving that the torsion on J1(N)(Q) is generated by Galois-orbits of cusps of X1(N) for N ≤55, N ̸=54.more » « less
-
Abstract We complete the computation of all
-rational points on all the 64 maximal Atkin-Lehner quotients$$\mathbb {Q}$$ 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 levels$$X_0(N)^*$$ N , we classify all -rational points as cusps, CM points (including their CM field and$$\mathbb {Q}$$ j -invariants) and exceptional ones. We further indicate how to use this to compute the -rational points on all of their modular coverings.$$\mathbb {Q}$$ -
Abstract We give new instances where Chabauty–Kim sets can be proved to be finite, by developing a notion of “generalised height functions” on Selmer varieties. We also explain how to compute these generalised heights in terms of iterated integrals and give the 1st explicit nonabelian Chabauty result for a curve $X/\mathbb{Q}$ whose Jacobian has Mordell–Weil rank larger than its genus.more » « less