Modular equations of hyperelliptic X₀(N) and an application

Hibino, Takeshi; Murabayashi, Naoki

doi:10.4064/aa-82-3-279-291

Cited by 10 publications

(14 citation statements)

References 10 publications

(3 reference statements)

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“…We encounter the same situation with the order-two operatorsω 11 (x),ω 17 (x),ω 19 (x),ω 29 (x),ω 31 (x),ω 41 (x),ω 47 (x), ω 59 (x), andω 71 (x), the corresponding two series at x = ∞ having no logarithmic terms yielding the same obstruction for a relation like (149). These various order-two operators correspond to higher order genus modular curves [153], namely genus-one forω 11 (x),ω 17 (x),ω 19 (x), genus-two forω 29 (x),ω 31 (x), genus-three forω 41 (x), genus-four forω 47 (x), genus-five forω 59 (x), and genus-six forω 71 (x). Note that all these higher-genusω n (x)'s are simply homomorphic to their adjoint.…”

Section: Appendix H2 More Detailsmentioning

confidence: 99%

Isingn-fold integrals as diagonals of rational functions and integrality of series expansions

Bostan¹,

Boukraa²,

Christol³

et al. 2013

J. Phys. A: Math. Theor.

361

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Abstract. We show that the n-fold integrals χ (n) of the magnetic susceptibility of the Ising model, as well as various other n-fold integrals of the "Ising class", or n-fold integrals from enumerative combinatorics, like lattice Green functions, correspond to a distinguished class of functions generalising algebraic functions: they are actually diagonals of rational functions. As a consequence, the power series expansions of the, analytic at x = 0, solutions of these linear differential equations "Derived From Geometry" are globally bounded, which means that, after just one rescaling of the expansion variable, they can be cast into series expansions with integer coefficients. We also give several results showing that the unique analytical solution of Calabi-Yau ODEs, and, more generally, Picard-Fuchs linear ODEs with solutions of maximal weights, are always diagonal of rational functions. Besides, in a more enumerative combinatorics context, generating functions whose coefficients are expressed in terms of nested sums of products of binomial terms can also be shown to be diagonals of rational functions. We finally address the question of the relations between the notion of integrality (series with integer coefficients, or, more generally, globally bounded series) and the modularity of ODEs.This paper is the short version of the larger (100 pages) version [19], available on http: // arxiv. org/ abs/ 1211. 6031 , where all the detailed proofs are given and where a larger set of examples is displayed.

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Section: Appendix H2 More Detailsmentioning

confidence: 99%

Isingn-fold integrals as diagonals of rational functions and integrality of series expansions

Bostan¹,

Boukraa²,

Christol³

et al. 2013

J. Phys. A: Math. Theor.

361

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show abstract

“…As an illustration, one can check that under this new form it readily gives triviality of X + 0 (37)(Q), for instance, whereas the previous version could not deal with this case and we had to invoke instead peculiar studies of level 37 by Hibino, Murabayashi, Momose and Shimura (cf. [16], [25]), as discussed in Section 6, page 9 of [26].…”

Section: Variant Of Mazur's Techniquesmentioning

confidence: 99%

Rational points on X_0^+ (p^r )

2013

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We show how the recent isogeny bounds due to Gaudron and Rémond allow to obtain the triviality of X + 0 (p r )(Q), for r > 1 and p a prime exceeding 2 · 10 11 . This includes the case of the curves X split (p). We then prove, with the help of computer calculations, that the same holds true for p in the range 11 ≤ p ≤ 10 14 , p = 13. The combination of those results completes the qualitative study of such sets of rational points undertook in [4] and [5], with the exception of p = 13.

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“…Proposition 3.12. Let x and f be as in Proposition 3.11 above, and let f + be the morphisms defined in Section 3.1 for a triple (p, r, s) in Table ( We also use the results in [16], [8] and [7].…”

Section: Local Moduli and Xmentioning

confidence: 99%

“…T. Hibino has given the relation of certain defining equation of X 0 (37) and invariant j-function. By using this result, T. Hibino and N. Murabayashi have decided the Qrational points of X split (37) ∼ = X + 0 (37 2 ) ( [8]). …”

Section: Local Moduli and Xmentioning

confidence: 99%

Lifting of supersingular points on X₀ (p^r) and lower bound of ramification index

Momose

Shimura

2002

Nagoya Mathematical Journal

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Abstract. Let K be a finite extension of Q ur p (= the maximal unramified extension of Qp) of degree eK, O its integer ring, p a rational prime and r a positive integer. If there exists a one parameter formal group defined over O whose reduction is of height 2 with a cyclic subgroup V of order p r defined over O, then eK ≥ 2p l (resp. p l + p l−1 ) if r = 2l + 1 (resp. r = 2l). We apply this result to a criterion for non-existence of Q-rational point of X + 0 (p r ). (This criterion is Momose's theorem in [14] except for the cases p = 5 and p = 13, but our new proof does not require defining equations of modular curves except for the case p = 2.) §0. Introduction Let p be a rational prime and let r be a positive integer. We denote Q p and Q ur p the p-adic number field and the maximal unramified extension of Q p . Let K be a finite extension of Q ur p , O the ring of integers of K, m the maximal ideal of O and e K the degree of K over Q ur p . Let E be an elliptic curve with cyclic subgroup A of order p r defined over O whose reduction mod m is a supersingular elliptic curve. In this paper, we will show that the existence of such a pair (E, A) gives a lower bound for e K with respect to p and r. If r is greater than one, the known lower bound of e K is p + 1 ([14]). We note that it depends only on p. Our main result is as follows.Main Theorem. Notation is as above. If such a pair (E, A) exists, thenWe have two proofs of this theorem, one is obtained by using the formal groups associated to elliptic curves, the other is due to the crossing theorem of modular curves ([9, Chap. 13, Theorem 13.4.7] This result is already proved by the first author, except for the cases p = 5 and p = 13. The result for p = 5 is based on the finiteness of the Q-rational points of J − 0 (125) ([16]). To prove this theorem, the first author used the defining equations of X 0 (p r ) in [14] for the cases p = 2, 3, 5. But our new proof requires the discussions of the defining equations only for the cases p = 2 with r = 6. Acknowledgement. We would like to thank the referee for one's many helpful remarks concerning our paper. In particular, Theorem 3.3 and its related lemmas are owed to the referee.

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Modular equations of hyperelliptic X₀(N) and an application

Cited by 10 publications

References 10 publications

Isingn-fold integrals as diagonals of rational functions and integrality of series expansions

Isingn-fold integrals as diagonals of rational functions and integrality of series expansions

Rational points on X_0^+ (p^r )

Lifting of supersingular points on X₀ (p^r) and lower bound of ramification index

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Modular equations of hyperelliptic X₀(N) and an application

Cited by 10 publications

References 10 publications

Isingn-fold integrals as diagonals of rational functions and integrality of series expansions

Isingn-fold integrals as diagonals of rational functions and integrality of series expansions

Rational points on X_0^+ (p^r )

Lifting of supersingular points on X0 (pr) and lower bound of ramification index

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Lifting of supersingular points on X₀ (p^r) and lower bound of ramification index