Hyperelliptic modular curves $X*_0(N)$

Hasegawa, Yuji

doi:10.4064/aa-81-4-369-385

Cited by 16 publications

(14 citation statements)

References 10 publications

(6 reference statements)

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“…In this case, we know that X * 0 (N ) is hyperelliptic for sixty-four values of N (cf. [Has97] ), and in [BG19] the following result is presented.…”

Section: Introductionmentioning

confidence: 87%

“…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. Also, it is useful when we find equations for the candidate non hyperelliptic curves X * 0 (N ) such that g * N > 4 (see Remark 2.7).…”

Section: Lemma 23 (Silverman-harrismentioning

confidence: 94%

“…Proof. By Table 3 in [Has97], we know that the curves X * 0 (207)/ V 3 and X * 0 (315)/ V 3 have genus two. By [Acc94], if X is a genus 3 curve which has a map to a genus 2 curve then X is hyperelliptic and bielliptic.…”

Section: Lemma 23 (Silverman-harrismentioning

confidence: 99%

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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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“…In this case, we know that X * 0 (N ) is hyperelliptic for sixty-four values of N (cf. [Has97] ), and in [BG19] the following result is presented.…”

Section: Introductionmentioning

confidence: 87%

Section: Lemma 23 (Silverman-harrismentioning

confidence: 94%

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Bielliptic modular curves X0⁎(N)

Bars

González

2020

Journal of Algebra

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“…Remark 2.4. In fact, the papers cited above together with [Has97] contain the information needed to list all (not necessarily prime) ∈ Z >0 with g(X * 0 ( )) = 0, 1, 2. 2.3.…”

Section: Lemmasmentioning

confidence: 99%

Unramified covers of Galois covers of low genus curves

Poonen

2005

Mathematical Research Letters

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Abstract. Let X → Y be a Galois covering of curves, where the genus of X is ≥ 2 and the genus of Y is ≤ 2. We prove that under certain hypotheses, X has an unramified cover that dominates a hyperelliptic curve; our results apply, for instance, to all tamely superelliptic curves. Combining this with a theorem of Bogomolov and Tschinkel shows that X has an unramified cover that dominates y 2 = x 6 − 1, if char k is not 2 or 3.

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“…Γ 0 (N)). Many papers have already been devoted to the problem of finding Q-rational models for these modular curves and their quotients [22,12,17,14,15,10]. In this work we are interested in modular curves defined over Q which are dominated over Q by X 1 (N).…”

Section: Introductionmentioning

confidence: 98%

Non-hyperelliptic modular curves of genus 3

González–Jiménez

Oyono

2010

Journal of Number Theory

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A curve C defined over Q is modular of level N if there exists a non-constant morphism from X 1 (N) onto C defined over Q for some positive integer N. We provide a sufficient and necessary condition for the existence of a modular non-hyperelliptic curve C of genus 3 and level N such that Jac C is Q-isogenous to a given three dimensional Q-quotient of J 1 (N). Using this criterion, we present an algorithm to compute explicitly equations for modular non-hyperelliptic curves of genus 3. Let C be a modular curve of level N, we say that C is new if the corresponding morphism between J 1 (N) and Jac C factors through the new part of J 1 (N).We compute equations of 44 non-hyperelliptic new modular curves of genus 3, that we conjecture to be the complete list of this kind of curves. Furthermore, we describe some aspects of nonnew modular curves and we present some examples that show the ambiguity of the non-new modular case.

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Hyperelliptic modular curves $X*_0(N)$

Cited by 16 publications

References 10 publications

Bielliptic modular curves X0⁎(N)

Bielliptic modular curves X0⁎(N)

Unramified covers of Galois covers of low genus curves

Non-hyperelliptic modular curves of genus 3

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