Trigonal Modular Curves $X_0^{+d}(N)$

Hasegawa, Yuji; Shimura, Mahoro

doi:10.3792/pjaa.75.172

Cited by 8 publications

(6 citation statements)

References 14 publications

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“…A series of works [37], [26], [25] led up to the determination of all 64 values of N for which the quotient of X 0 (N) by its Atkin-Lehner group, X * (N), is hyperelliptic, and this was generalized in [19] to quotients of X 0 (N) by an arbitrary subgroup of the Atkin-Lehner group. Similar results determining all trigonal curves of the form X 0 (N), X 0 (N)/W d for a single Atkin-Lehner involution W d , and X * (N), can be found in [27], [28], and [29], respectively. Some of these curves are not new, and hence do not appear in our tables.…”

Section: 23supporting

confidence: 77%

Finiteness results for modular curves of genus at least 2

Baker¹,

González–Jiménez²,

González³

et al. 2005

ajm

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A curve X over Q is modular if it is dominated by X 1 (N ) for some N ; if in addition the image of its jacobian in J 1 (N ) is contained in the new subvariety of J 1 (N ), then X is called a new modular curve. We prove that for each g ≥ 2, the set of new modular curves over Q of genus g is finite and computable. For the computability result, we prove an algorithmic version of the de Franchis-Severi Theorem. Similar finiteness results are proved for new modular curves of bounded gonality, for new modular curves whose jacobian is a quotient of J 0 (N ) new with N divisible by a prescribed prime, and for modular curves (new or not) with levels in a restricted set. We study new modular hyperelliptic curves in detail. In particular, we find all new modular curves of genus 2 explicitly, and construct what might be the complete list of all new modular hyperelliptic curves of all genera. Finally we prove that for each field k of characteristic zero and g ≥ 2, the set of genus-g curves over k dominated by a Fermat curve is finite and computable. Curves and varieties in this paper are smooth, projective, and geometrically integral, unless otherwise specified. When we write an affine equation for a curve, its smooth projective model is implied.1 Question 1.12. Is it true that for every field k of characteristic zero, and every g ≥ 2, the set of k-modular curves over k of genus g up to k-isomorphism is finite?Remark 1.13. If X is a k-modular curve over k, and we define k 0 = k ∩ Q ⊆ k, then X = X 0 × k 0 k for some k 0 -modular curve X 0 . This follows from the de Franchis-Severi Theorem.Remark 1.14. If k and k ′ are fields of characteristic zero with [k ′ : k] finite, then a positive answer to Question 1.12 for k ′ implies a positive answer for k, since Galois cohomology and the finiteness of automorphism group of curves of genus at least 2 show that for each X ′ over k ′ , there are at most finitely many curves X over k with X × k k ′ ≃ X ′ . But it is not clear, for instance, that a positive answer for Q implies or is implied by a positive answer for Q. Recovering curve information from differentials2.1. Recovering a curve from partial expansions of its differentials. The goal of this section is to prove the following result, which will be used frequently in the rest of this paper.Proposition 2.1. Fix an integer g ≥ 2. There exists an integer B > 0 depending on g such that if k is a field of characteristic zero, and w 1 , . . . , w g are elements of k[[q]]/(q B ), then up to k-isomorphism, there exists at most one curve X over k such that there exist P ∈ X(k) and an analytic uniformizing parameter q in the completed local ringÔ X,P such that w 1 dq, . . . , w g dq are the expansions modulo q B of some basis of H 0 (X, Ω).Lemma 2.2. Let k be a field, and let X/k be a curve of genus g. Let P ∈ X(k) be a k-rational point, let F ∈ k[t 1 , . . . , t g ] be a homogeneous polynomial of degree d, and let q be an analytic uniformizing parameter inÔ X,P . Suppose we are given elements ω 1 , . . . , ω g ∈ H 0 (X, Ω), and for each i...

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Section: 23supporting

confidence: 77%

Finiteness results for modular curves of genus at least 2

Baker¹,

González–Jiménez²,

González³

et al. 2005

ajm

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“…Example Let us carry out the foregoing procedure for the curve defined by (x 3 + x + 1)y 3 + 42(2x 4 + x 3 + 3x 2 + 3x + 1)y 2 + (x + 1)(x 4 + 2x 2 + x + 1)y + 42(x 2 + 1) = 0 over F 43 . This is the reduction mod 43 of the modular curve X + 0 (164), or rather an affine model of it, whose equation we took from [32]. It is of genus 6, while we note that Baker's bound reads 7, so it is not met here.…”

Section: Methodsmentioning

confidence: 99%

Point counting on curves using a gonality preserving lift

Castryck

Tuitman

2017

The Quarterly Journal of Mathematics

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We study the problem of lifting curves from finite fields to number fields in a genus and gonality preserving way. More precisely, we sketch how this can be done efficiently for curves of gonality at most four, with an in-depth treatment of curves of genus at most five over finite fields of odd characteristic, including an implementation in Magma. We then use such a lift as input to an algorithm due to the second author for computing zeta functions of curves over finite fields using p-adic cohomology.

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“…Proof. On the one hand, combining [37, Theorem 3] (see also [24,Theorem 4.3]) and a remark in [16], the index of a congruence subgroup in PSL 2 (Z) is bounded by 101 times the gonality of the corresponding modular curve. By assumption the modular curves admits a smooth plane model, so its gonality is d − 1.…”

Section: A Bound For the Genusmentioning

confidence: 99%

“…The equation of the model is the minimal polynomial of the modular function ∆(N z)/∆ over C(j), where ∆ is the Ramanujan ∆ function and j is the modular j function. Some plane (singular) models for modular curves X 0 (N ) were already found by Hasegawa and Shimura in [24] using different ideas, in particular studying the gonality of modular curves.…”

Section: Introductionmentioning

confidence: 96%