Explicit Chabauty–Kim for the split Cartan modular curve of level 13

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

doi:10.4007/annals.2019.189.3.6

Cited by 70 publications

(128 citation statements)

References 38 publications

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“…For N ∈ {67, 73}, it then remains to determine the rational points on X 0 (N ) + , which is hyperelliptic of genus 2 and rank 2 in both cases. These curves X 0 (67) + and X 0 (73) + turn out to be beautiful test cases for the explicit quadratic Chabauty method developed by Balakrishnan and Dogra [2] and Balakrishnan, Dogra, Müller, Tuitman and Vonk [3] following Kim's work [20], [21] on nonabelian Chabauty. The author understands that Balakrishnan, Best, Bianchi, Müller, Triantafillou and Vonk are working on applying quadratic Chabauty to these and more curves, and a proof for X 0 (67) + and X 0 (73) + by (a subset of) these authors is expected to appear in the near future.…”

Section: Introductionmentioning

confidence: 93%

Quadratic points on modular curves with infinite Mordell–Weil group

Box¹

2020

Math. Comp.

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Bruin and Najman [5] and Ozman and Siksek [33] have recently determined the quadratic points on each modular curve X 0 (N ) of genus 2, 3, 4, or 5 whose Mordell-Weil group has rank 0. In this paper we do the same for the X 0 (N ) of genus 2, 3, 4, and 5 and positive Mordell-Weil rank. The values of N are 37, 43, 53, 61, 57, 65, 67 and 73. The main tool used is a relative symmetric Chabauty method, in combination with the Mordell-Weil sieve. Often the quadratic points are not finite, as the degree 2 map X 0 (N ) → X 0 (N ) + can be a source of infinitely many such points. In such cases, we describe this map and the rational points on X 0 (N ) + , and we specify the exceptional quadratic points on X 0 (N ) not coming from X 0 (N ) + . In particular we determine the j-invariants of the corresponding elliptic curves and whether they are Q-curves or have complex multiplication. 1 2 JOSHA BOX for N ≥ B d are cuspidal. Moreover, for prime values of N , Merel's uniform boundedness theorem [30] proves the existence of such an upper bound B d for each d.The low degree points on X 0 (N ), however, are naturally more abundant than on X 1 (N ), which complicates their study.There are two obvious potential sources of infinitely many quadratic points on X 0 (N ): a degree 2 map X 0 (N ) → P 1 over Q (which exists if and only if X 0 (N ) is hyperelliptic) and a degree 2 map over Q to an elliptic curve with infinite Mordell-Weil group (the existence of which implies that X 0 (N ) is bielliptic). In both of these cases the rational points on the image give rise to infinitely many quadratic points on X 0 (N ). Using Faltings' theorem [13] on abelian varieties, Abramovich and Harris [1] show that the set of quadratic points on X 0 (N ) is infinite in these two cases only. Building on the work of Harris and Silverman [15], Bars [4] then determined all bielliptic X 0 (N ) and decided that exactly 10 of these have a quotient of positive Mordell-Weil rank. Moreover, Ogg [31] decided for which 19 values of N the curve X 0 (N ) is hyperelliptic. Consequently, X 0 (N ) has infinitely many quadratic points for the following 28 explicit values of N 131. The careful reader will have noticed there must be one curve that is both hyperelliptic and bielliptic with quotient of positive rank: this is the infamous X 0 (37).Even when infinite, the quadratic points on X 0 (N ) can be described. Recently, Bruin and Najman [5] determined the finitely many exceptional quadratic points (those not coming from P 1 (Q)) on all hyperelliptic X 0 (N ) except for N = 37, and they moreover proved in those cases that the quadratic points coming from P 1 (Q) correspond to Q-curves. The remaining case N = 37 is also the only hyperelliptic one where J 0 (N )(Q) is infinite. Subsequently, Ozman and Siksek [33] determined all finitely many quadratic points on X 0 (N ) when it is non-hyperelliptic of genus 2, 3, 4 or 5 and the Mordell-Weil group J 0 (N )(Q) is finite.

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Section: Introductionmentioning

confidence: 93%

Quadratic points on modular curves with infinite Mordell–Weil group

Box¹

2020

Math. Comp.

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“…(B) For the following exceptional types there are only a finite number of Q-isomorphic classes: ⋆ 3Ns and G p is 5B. Moreover, we give unconditionally the moduli space for each of the possible exceptional types (see Tables 1,2,3,6,7,9,11,12), except for the cases of level 13, 17 and 37 which are conditionally under Conjecture 3 and the Strong Uniformity Conjecture. Corollary 18.…”

Section: Theorem 17 (A) the Following Nontrivial Exceptional Types Omentioning

confidence: 99%

“…⋆ [8X5,3Nn]: Let E be the modular curve associated to [8X5,3Nn]. In this case, E/Q is the elliptic curve with Cremona label 576a3 and has j-map equal to j(x) = 8x 3 where (x, y) ∈ E(Q). Now let 8X17 be the group…”

Section: Proof Of the Theorem 17 Theorem 19 Corollary 18 And Coromentioning

confidence: 99%

“…Let E/Q be a non-CM elliptic curve and p a prime. Then one the following possibilities occurs: 3,5,7,11,13,17, 37}, and G E (p) is conjugate in GL 2 (F p ) to one of the groups in Tables 1 and 2. (iii) p ≥ 17: G E (p) is conjugate to a subgroup of the normalizer of a non-split Cartan subgroup of level p. Definition 6. Let E/Q be a non-CM elliptic curve, then we define Serre's constant associated to E to be…”

Section: Introductionmentioning

confidence: 99%

“…Theorem 10. Assuming the Uniformity Conjecture: 3,4,5,6,7,8,9,10,11,12,13,14,15,20,21,24,28,40, 56, 104}.…”

Section: Introductionmentioning

confidence: 99%

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Serre’s Constant of Elliptic Curves Over the Rationals

Daniels

González–Jiménez

2019

Experimental Mathematics

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Let E be an elliptic curve without complex multiplication defined over the rationals. The purpose of this article is to define a positive integer A(E), that we call the Serre's constant associated to E, that gives necessary conditions to conclude that ρE,m, the mod m Galois representation associated to E, is non-surjective. In particular, if there exists a prime factor p of m satisfying valp(m) ≥ valp(A(E)) > 0 then ρE,m is non-surjective. Conditionally under Serre's Uniformity Conjecture, we determine all the Serre's constants of elliptic curves without complex multiplication over the rationals that occur infinitely often. Moreover, we give all the possible combination of mod p Galois representations that occur for infinitely many non-isomorphic classes of non-CM elliptic curves over Q, and the known cases that appear only finitely. We obtain similar results for the possible combination of maximal non-surjective subgroups of GL2(Zp). Finally, we conjecture all the possibilities of these combinations and in particular all the possibilities of these Serre's constants.Serre's Uniformity Question. If E/Q is a non-CM elliptic curve, then must it be that ρ E,p is surjective for any prime p ≥ 41?Nowadays, an affirmative answer to the above question has received the name of Serre's Uniformity Conjecture (or sometimes just Uniformity Conjecture) despite the fact that Serre himself never conjectured it to be true.

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Square Root Time Coleman Integration on Superelliptic Curves

Best

2021

Arithmetic Geometry, Number Theory, and Computation

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Explicit Chabauty–Kim for the split Cartan modular curve of level 13

Cited by 70 publications

References 38 publications

Quadratic points on modular curves with infinite Mordell–Weil group

Quadratic points on modular curves with infinite Mordell–Weil group

Serre’s Constant of Elliptic Curves Over the Rationals

Square Root Time Coleman Integration on Superelliptic Curves

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