Quadratic Chabauty for modular curves: Algorithms and examples

Balakrishnan, Jennifer S.; Dogra, Netan; Müller, J. Steffen; Tuitman, Jan; Vonk, J. Arie

doi:10.48550/arxiv.2101.01862

Cited by 3 publications

(12 citation statements)

References 30 publications

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“…(2) Since the determinant of E;p is the mod p cyclotomic character, we have thatP E;p .G K / Â PSL 2 .F p / if and only if p p 2 K.When this is the case, by Lemma 2.7, we see that each index 2 subgroup of the projective image is split when p Á 1 .mod 4/ and non-split when p Á 3 .mod 4/. By part(1), it follows in both cases that the congruences between E and each of the three twists are all symplectic. Now suppose that p p … K. By Lemma 2.7 again, exactly one of the three subgroups is split when p Á 1 .mod 4/, and exactly one is non-split when p Á 3 .mod 4/, so in both cases exactly one congruence is symplectic.…”

mentioning

confidence: 85%

“…In Corollary 1.9 of [2] there is a list of 3 rational j -invariants of elliptic curves over Q satisfying (iv); it has recently been shown (see [1]) that the associated genus 3 modular curve X S 4 .13/ found explicitly in [2] has no more rational points, and hence that this list is complete. By Theorem 2.4, none of these curves is mod 13 congruent to a twist of another of them (including itself); the same is true for the curves and values of p in case (v).…”

Section: Further Motivationmentioning

confidence: 99%

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Global methods for the symplectic type of congruences between elliptic curves

Cremona

Freitas

2021

Rev. Mat. Iberoam.

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We describe a systematic investigation into the existence of congruences between the mod p torsion modules of elliptic curves defined over Q, including methods to determine the symplectic type of such congruences. We classify the existence and symplectic type of mod p congruences between twisted elliptic curves over number fields, giving global symplectic criteria that apply in situations where the available local methods may fail. We report on the results of applying our methods for all primes p 7 to the elliptic curves in the LMFDB database, which currently includes all elliptic curves of conductor less than 500 000. We also show that while such congruences exist for each p Ä 17, there are none for p 19 in the database, in line with a strong form of the Frey-Mazur conjecture.

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mentioning

confidence: 85%

Section: Further Motivationmentioning

confidence: 99%

Global methods for the symplectic type of congruences between elliptic curves

Cremona

Freitas

2021

Rev. Mat. Iberoam.

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“…They also compute the Q-curves corresponding to each non-cuspidal point in the set. The prime levels N = 67, 73, 103, 107, 161, 167, 191 were approached using quadratic Chabauty in [5] and [11]. We compute X 0 (N ) * (Q) for the remaining square-free levels…”

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

confidence: 99%

“…When N is a prime power, X 0 (N ) * = X 0 (N ) + , since w N is the only non-trivial Atkin-Lehner involution. For N prime and genus(X 0 (N ) * ) ∈ {2, 3}, the sets X 0 (N ) * (Q) were computed in [5,11]. 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].…”

Section: Introductionmentioning

confidence: 99%

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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“…5,4). Hence the model (13.1) of E a,b is minimal at 2 and E a,b has additive potential good reduction at 2.…”

mentioning

confidence: 96%

On Darmon's program for the Generalized Fermat equation

Billerey¹,

Chen²,

Dieulefait³

et al. 2022

Preprint

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In 2000, Darmon described a program to study the generalized Fermat equation using modularity of abelian varieties of GL 2 -type over totally real fields. The original program was based on hard open conjectures, which has made it difficult to apply in practice.In this paper, building on the progress surrounding the modular method from the last two decades, we analyze and expand the current limits of this program by developing all the necessary ingredients to use Frey abelian varieties for two Diophantine applications.In the first application, we use a multi-Frey approach combining two Frey elliptic curves over totally real fields, a Frey hyperelliptic over Q due to Kraus, and ideas from the Darmon program to give a complete resolution of the generalized Fermat equationfor all integers n ≥ 2. The use of higher dimensional Frey abelian varieties allows a more efficient proof of the above result due to additional structures that they afford.In the second application, we use some of the additional structures that Frey abelian varieties possess to give an asymptotic resolution of the generalized Fermat equationx 11 + y 11 = z p for solutions locally away from xy = 0 and where p is a prime exponent. In this application, the use of higher dimensional Frey abelian varieties helps to overcome the computational difficulties arising when working in large spaces of Hilbert modular forms.

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Quadratic Chabauty for modular curves: Algorithms and examples

Cited by 3 publications

References 30 publications

Global methods for the symplectic type of congruences between elliptic curves

Global methods for the symplectic type of congruences between elliptic curves

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

On Darmon's program for the Generalized Fermat equation

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