Trigonal quotients of modular curves $X_{0}(N)$

Hasegawa, Yuji; Shimura, Mahoro

doi:10.3792/pjaa.82.15

Cited by 6 publications

(8 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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“…Therefore if X * 0 (N ) is bielliptic and non-hyperelliptic with g * N = 3 or 4, then it has gonality three and is a trigonal curve. Hasegawa and Shimura listed all trigonal modular curves X * 0 (N ) in [HS00]. This result (jointly with [Has97]) allows us to present the complete list of the values N such that 3 ≤ g * N ≤ 4 in Appendix A.…”

Section: Lemma 23 (Silverman-harrismentioning

confidence: 95%

“…For all these values, the curve X * 0 (N ) is neither trigonal (cf. [HS00]) nor a smooth plane quintic. Hence, if ω j is the regular differential of an elliptic curve E, the pair (X * 0 (N ), E) is bielliptic over Q if, and only if, dim L 2,j = 3.…”

Section: *mentioning

confidence: 98%

Bielliptic modular curves X0⁎(N)

Bars

González

2020

Journal of Algebra

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Let N ≥ 1 be a integer such that the modular curve X * 0 (N) has genus ≥ 2. We prove that X * 0 (N) is bielliptic exactly for 69 values of N. In particular, we obtain that X * 0 (N) is bielliptic over the base field for all these values of N , except X * 0 (160) that is not bielliptic over Q but it does over Q(√ −1). Moreover, we prove that the set of all quadratic points over Q for the modular curve X * 0 (N) is infinite exactly for 100 values of N .

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“…Only X * 0 (370) could be bielliptic. In this case, the curve is trigonal (see [16,Proposition 1]) and dim L 3 = 5. By computing a polynomial Q 3 ∈ L 3 that is not multiple of Q 2 , we get…”

Section: Even Casementioning

confidence: 99%

Bielliptic modular curves $X_0^*(N)$ with square-free levels

Bars

Rovira

2019

Math. Comp.

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Let N ≥ 1 be a square-free integer such that the modular curve X * 0 (N ) has genus ≥ 2. We prove that X * 0 (N ) is bielliptic exactly for 19 values of N , and we determine the automorphism group of these bielliptic curves. In particular, we obtain the first examples of nontrivial Aut(X * 0 (N )) when the genus of X * 0 (N ) is ≥ 3. Moreover, we prove that the set of all quadratic points over Q for the modular curve X * 0 (N ) with genus ≥ 2 and N square-free is not finite exactly for 51 values of N .

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Trigonal quotients of modular curves $X_{0}(N)$

Cited by 6 publications

References 14 publications

Finiteness results for modular curves of genus at least 2

Finiteness results for modular curves of genus at least 2

Bielliptic modular curves X0⁎(N)

Bielliptic modular curves $X_0^*(N)$ with square-free levels

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