Rational points on $X^+_0(125)$

Arul, Vishal; Müller, J. Steffen

doi:10.48550/arxiv.2205.14744

Cited by 1 publication

(7 citation statements)

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“…Thus, this completes the determination of rational points on all hyperelliptic curves X 0 (N ) * . Together with the results in [1,4,5,11,13,18], we have the following theorem.…”

Section: Theorem 12 ([34 Theorem B])supporting

confidence: 63%

“…For the square-free levels we also compute the j-invariants of the Q-curves associated to non-cuspidal rational points. The methods of [13], quadratic and bielliptic quadratic Chabauty, and recent work of Arul and Müller [4] together determine X 0 (N ) * (Q) for all of the 18 non-squarefree levels N as explained in Sect. 3.2.…”

Section: Theorem 12 ([34 Theorem B])mentioning

confidence: 99%

“…Remark 1. 4 The boundedness conjecture of Elkies [26] predicts that there are only finitely many positive integers N such that X 0 (N ) * contains an exceptional rational point. Although the main theorem of this paper says that exceptional rational points exist on most of the hyperelliptic curves X 0 (N ) * of squarefree level N , we observe from Table 6 that most of these exceptional points arise as the image of a cusp or CM point under the hyperelliptic involution.…”

Section: Main Theoremmentioning

confidence: 99%

“…(For the definition of a twist of a finite étale Galois covering, see [55, §4].) Note that these are finite étale coverings with geometric Galois group (Z/2Z) 4 . One has…”

Section: Elliptic Curve Chabautymentioning

confidence: 99%

“…For N prime and genus(X 0 (N ) * ) ∈ {4, 5, 6}, the same was carried out by the first three authors and Vishal Arul, Lea Beneish, Mingjie Chen, and Boya Wen in [1]. For N composite and genus(X 0 (N ) + ) ∈ {2, 3, 4, 5, 6}, the articles [3,4,31,47] complete the determination of the sets X 0 (N ) + (Q). These articles together verify a conjecture of Galbraith [31] on the set of exceptional rational points on the curves X 0 (N ) + with genus between 2 and 5.…”

Section: Introductionmentioning

confidence: 99%

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Rational points on hyperelliptic Atkin-Lehner quotients of modular curves and their coverings

et al. 2022

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We complete the computation of all $$\mathbb {Q}$$ Q -rational points on all the 64 maximal Atkin-Lehner quotients $$X_0(N)^*$$ 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 levels N, we classify all $$\mathbb {Q}$$ Q -rational points as cusps, CM points (including their CM field and j-invariants) and exceptional ones. We further indicate how to use this to compute the $$\mathbb {Q}$$ Q -rational points on all of their modular coverings.

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“…Thus, this completes the determination of rational points on all hyperelliptic curves X 0 (N ) * . Together with the results in [1,4,5,11,13,18], we have the following theorem.…”

Section: Theorem 12 ([34 Theorem B])supporting

confidence: 63%

Section: Theorem 12 ([34 Theorem B])mentioning

confidence: 99%

Section: Main Theoremmentioning

confidence: 99%

“…(For the definition of a twist of a finite étale Galois covering, see [55, §4].) Note that these are finite étale coverings with geometric Galois group (Z/2Z) 4 . One has…”

Section: Elliptic Curve Chabautymentioning

confidence: 99%