The kernel of Ribet's isogeny for genus three Shimura curves

González, Josep; Molina, Santiago

doi:10.2969/jmsj/06820609

Cited by 8 publications

(14 citation statements)

References 16 publications

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“…Now the quadratic form associated to Y ′ is 4x 2 + 5y 2 . Considering matrices of the form We can deduce from (14) and (15) that our result agrees with (2).…”

Section: Modular Parameterization Of Shimura Curves On Asupporting

confidence: 73%

“…(The same equation also appeared in [14], which in turn was obtained from an earlier work of Molina [26] by a simple change of variables.) Since the covering X → X/w 13 ramifies at CM-points of discriminant −52, we find that x takes values ±i at the two CM-points of discriminant −52.…”

Section: Evaluations Of Modular Functionsmentioning

confidence: 55%

“…where m is an integer such that r 2 m is representable by the quadratic form associated to X for some rational number r. For instance, the quadratic form for X 2 51 is 5x 2 + 2xy + 41y 2 , which clearly represents 5. Also, a canoncal model for X 51 0 (1) is y 2 = f (x), where f (x) = −(x 2 + 3)(243x 6 + 235x 4 − 31x 2 + 1) (see [14,15]). Our computation shows that the Mestre obstruction for X 2 51 is −51,5f (j) Q(j)…”

Section: Q(j)mentioning

confidence: 99%

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Quaternionic loci in Siegel’s modular threefold

Lin

Yang

2019

Math. Z.

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Let Q D be the set of moduli points on Siegel's modular threefold whose corresponding principally polarized abelian surfaces have quaternionic multiplication by a maximal order O in an indefinite quaternion algebra of discriminant D over Q such that the Rosati involution coincides with a positive involution of the form α → µ −1 αµ on O for some µ ∈ O with µ 2 + D = 0. In this paper, we first give a formula for the number of irreducible components in Q D , strengthening an earlier result of Rotger. Then for each irreducible component of genus 0, we determine its rational parameterization in terms of a Hauptmodul of the associated Shimura curve.

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“…Now the quadratic form associated to Y ′ is 4x 2 + 5y 2 . Considering matrices of the form We can deduce from (14) and (15) that our result agrees with (2).…”

Section: Modular Parameterization Of Shimura Curves On Asupporting

confidence: 73%

Section: Evaluations Of Modular Functionsmentioning

confidence: 55%

Section: Q(j)mentioning

confidence: 99%

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Quaternionic loci in Siegel’s modular threefold

Lin

Yang

2019

Math. Z.

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“…In this case, the proof is based on the fact that X N is bielliptic and the lattices of J 0 (N ) new and J N can be computed through their elliptic quotients.When dim(J N ) = 3, equiv. N = 2 · 31, 2 · 41, 2 · 47, 3 · 13, 3 · 17, 3 · 19, 3 · 23, 5 · 7, 5 · 11, Ogg's conjecture is verified in [6]. In this case, X N is always hyperelliptic.…”

mentioning

confidence: 79%

“…When dim(J N ) = 3, equiv. N = 2 · 31, 2 · 41, 2 · 47, 3 · 13, 3 · 17, 3 · 19, 3 · 23, 5 · 7, 5 · 11, Ogg's conjecture is verified in [6]. In this case, X N is always hyperelliptic.…”

mentioning

confidence: 79%

On Ribet’s isogeny for $J_0(65)$

Klosin

Papikian

2018

Proc. Amer. Math. Soc.

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Let J 65 be the Jacobian of the Shimura curve attached to the indefinite quaternion algebra over Q of discriminant 65. We study the isogenies J 0 (65) → J 65 defined over Q, whose existence was proved by Ribet. We prove that there is an isogeny whose kernel is supported on the Eisenstein maximal ideals of the Hecke algebra acting on J 0 (65), and moreover the odd part of the kernel is generated by a cuspidal divisor of order 7, as is predicted by a conjecture of Ogg.2010 Mathematics Subject Classification. 11G18. 1 2 KRZYSZTOF KLOSIN AND MIHRAN PAPIKIAN the isogenies between them. Ogg's conjecture remains open except for the special cases when J N has dimension ≤ 3.When dim(J N ) = 1, equiv. N = 2 · 7, 3 · 5, 3 · 7, 3 · 11, 2 · 17, J N is an elliptic curve over Q which is uniquely determined by its component groups at p and q, and J 0 (N ) new is the optimal elliptic curve of conductor N . Then one easily checks Ogg's conjecture using Cremona's tables [5]. In general, the orders of component groups of J N can be computed using Brandt matrices [10], which is relatively easy to do with the help of a computer program such as Magma.When dim(J N ) = 2, equiv. N = 2 · 13, 2 · 19, 2 · 29, Ogg's conjecture is verified in [7]. In this case, the proof is based on the fact that X N is bielliptic and the lattices of J 0 (N ) new and J N can be computed through their elliptic quotients.When dim(J N ) = 3, equiv. N = 2 · 31, 2 · 41, 2 · 47, 3 · 13, 3 · 17, 3 · 19, 3 · 23, 5 · 7, 5 · 11, Ogg's conjecture is verified in [6]. In this case, X N is always hyperelliptic. By utilizing this fact, González and Molina explicitly compute the equation for each X N . Then they obtain a basis of regular differentials for X N from these equations to produce a period matrix for J N . The period matrix of J 0 (N ) new can be computed using cusp forms with rational q-expansions. The problem then reduces to comparing the period matrices of appropriate quotients of J 0 (N ) new with the period matrix of J N .The goal of this paper is to study Ribet's isogeny for N = 5 · 13 = 65. In this case, dim(J N ) = 5 and X N is not hyperelliptic; cf. [14]. Our approach to the study of Ribet isogenies is completely different from that in [7] and [6], and crucially relies on the Hecke equivariance of such isogenies. In this approach we need to know very little about X N or J N ; we only need to know the orders of component groups of J N , which, as we mentioned, are easy to compute, and in fact were already computed in [16]. The difficulty shifts to the study of the structure of the Hecke algebra and its action on J 0 (N ).Let T(N ) := Z[T 2 , T 3 , . . . ] be the Z-algebra generated by the Hecke operators T n acting on be the space S 2 (N ) of weight 2 cups forms on Γ 0 (N ). This algebra is isomorphic to the subalgebra of End(J 0 (N )) generated by T n acting as correspondences on X 0 (N ). When N = 65, we have J 0 (N ) new = J 0 (N ), so there is a Ribet isogeny π : J 0 (N ) → J N . T(N ) also naturally acts on J N and π is T(N )-equivariant. This equivari...

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Rational points on Atkin–Lehner quotients of geometrically hyperelliptic Shimura curves

Oana

Schembri

2023

Expositiones Mathematicae

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The kernel of Ribet's isogeny for genus three Shimura curves

Cited by 8 publications

References 16 publications

Quaternionic loci in Siegel’s modular threefold

Quaternionic loci in Siegel’s modular threefold

On Ribet’s isogeny for $J_0(65)$

Rational points on Atkin–Lehner quotients of geometrically hyperelliptic Shimura curves

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