Defining Equations of Modular Curves $X_0(N)$

Shimura, Mahoro

doi:10.3836/tjm/1270043475

Cited by 36 publications

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“…The cases N = 8, 9 can in principle also be handled by Theorem 5.3, like N = 6, 7, since equations for the curves X 0 (64), X 0 (81) are known [29]. (X 0 (64), X 0 (81) are non-hyperelliptic of genera 3, 4, and X 0 (64) is the degree-4 Fermat curve.)…”

Section: Key Resultsmentioning

confidence: 99%

Algebraic hypergeometric transformations of modular origin

Maier

2007

Trans. Amer. Math. Soc.

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Abstract. It is shown that Ramanujan's cubic transformation of the Gauss hypergeometric function 2 F 1 arises from a relation between modular curves, namely the covering of X 0 (3) by X 0 (9). In general, when 2 N 7, the N -fold cover of X 0 (N ) by X 0 (N 2 ) gives rise to an algebraic hypergeometric transformation. The N = 2, 3, 4 transformations are arithmetic-geometric mean iterations, but the N = 5, 6, 7 transformations are new. In the final two cases the change of variables is not parametrized by rational functions, since X 0 (6), X 0 (7) are of genus 1. Since their quotients X + 0 (6), X + 0 (7) under the Fricke involution (an Atkin-Lehner involution) are of genus 0, the parametrization is by two-valued algebraic functions. The resulting hypergeometric transformations are closely related to the two-valued modular equations of Fricke and H. Cohn.

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Section: Key Resultsmentioning

confidence: 99%

Algebraic hypergeometric transformations of modular origin

Maier

2007

Trans. Amer. Math. Soc.

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“…While our equation confirms the hyperellipticity, it has (affine) singularities, which is reflected by the fact that its degree in X exceeds 8. Furthermore, its coefficients are considerably larger than those of the model in [12,16]. This kind of behaviour appears to be un-avoidable as long as we choose j as separating variable, which is important for applications in which one wishes to relate the modular polynomial to concrete equations of elliptic curves (e.g., the determination of the number of rational isogenies or the construction of elliptic curves with complex multiplication [4]).…”

Section: Examples and Conclusionmentioning

confidence: 98%

“…Topics of interest include the search for models with few or no singularities or with small coefficients. For instance, the question which curves X 0 (N ) are hyperelliptic has been answered in [11] and complemented by concrete models in [12,9,16]. The factorisation pattern of these modular equations provides information on the rationality of isogenies and can thus be used to determine the L-function of elliptic curves over finite fields.…”

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confidence: 99%

Modular curves of composite level

Enge

Schertz

2005

Acta Arith.

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1. Motivation. The modular curves X 0 (N ) have been studied intensively since they provide a link between modular and elliptic functions. Among other things, they parameterise pairs of elliptic curves with a cyclic isogeny of degree N between them. Topics of interest include the search for models with few or no singularities or with small coefficients. For instance, the question which curves X 0 (N ) are hyperelliptic has been answered in [11] and complemented by concrete models in [12,9,16]. The factorisation pattern of these modular equations provides information on the rationality of isogenies and can thus be used to determine the L-function of elliptic curves over finite fields. In general, plane equations with nice properties are obtained by looking for two functions on X 0 (N ) with suitable pole orders and determining a polynomial relationship between them. To link the modular equations to the pairs of isogenous elliptic curves they parameterise, one needs to exhibit a relationship with the modular invariant j, which requires considerable work in each case (see [5,2]).It is thus convenient to fix one of the functions as j and to consider X 0 (N ) as a cover of X 0 (1), albeit this introduces further singularities. Equations for X 0 (N )/X 0 (1) with small coefficients have been exhibited in [7] and [8, Chapter 5]; some ideas presented there go back to Atkin (see [2]). In the context of point counting on elliptic curves, it is most efficient to use curves of prime level N .In this article we deal with an infinite family of modular curves X 0 (N ) where N is composite of the simplest form, i.e. a product of two primes p 1 and p 2 , which need not be distinct. Besides j, we also fix the second function generating the function field of X 0 (N ) as a certain product of η-functions. This is motivated by the observation that the singular values of these functions at ideals of suitable orders in imaginary-quadratic number fields lie in the corresponding Hilbert and ring class fields [13]. The modular polynomials relating these functions and j can thus be used to explicitly determine

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“…Furthermore, arithmetic aspects of the theory can be found in a well-known book of Shimura ( [13]). The uniformization of modular curves is used to compute equations of modular curves X 0 (N ) in [2,6,14,16].…”

Section: Introductionmentioning

confidence: 99%

Birational Maps of X(1) into P2

Mikoć¹,

Muić²

2013

Glas. Mat. Ser. III

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Abstract. In this paper we study birational maps of modular curve X(1) attached to SL 2 (Z) into the projective plain P 2 . We prove that every curve of genus 0 and degree q in P 2 can be uniformized by modular forms for SL 2 (Z) of weight 12q but not with modular forms of smaller weight, and that the corresponding uniformization can be chosen to be a birational equivalence. We study other regular maps X(1) −→ P 2 and we compute the equation of obtained projective curve. We provide numerical examples in SAGE.

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Defining Equations of Modular Curves $X_0(N)$

Cited by 36 publications

References 0 publications

Algebraic hypergeometric transformations of modular origin

Algebraic hypergeometric transformations of modular origin

Modular curves of composite level

Birational Maps of X(1) into P2

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