Modular curves of composite level

Abstract

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

Constructing elliptic curves of prime order

“…The value of the double η-quotient f = η(z/5)η(z/13) η(z)η(z/65) at z = −21+ √ −2419 2 generates the Hilbert class field H −2419 . The minimal polynomial Ψ of f over C(j) can be computed as in [10]. It has degree 4 in j and degree 84 in X, and we have r(f ) = 84/4 = 21.…”

Modular equations for some η-products

“…Note that s | 24; and s | 6 when p 1 and p 2 are odd primes. It is shown in [6] that the function w s p 1 ,p 2 is a function on Γ 0 (p 1 p 2 ); properties of the classical modular equation are also given.…”

“…In [6] are given the conjugates of w s p 1 ,p 2 (with some minor typos). Here, we need precise the expansions of w p 1 ,p 2 .…”

Singular values of multiple eta-quotients for ramified primes

“…For a few functions of low level, however, the degree d j of Ψ f in j is exactly 2 k−1 , and no factorisation is needed for a discriminant for which all primes dividing N are ramified. For k = 2, a degree d j = 2 is obtained if and only if (p 1 − 1)(p 2 − 1) | 24 by [7,Theorem 9], that is, for {p 1 , p 2 } ∈ {{2, 3}, {2, 5}, {2, 7}, {2, 13}, {3, 5}, {3, 7}, {3, 13}, {5, 7}}. Conjecturally, for k 3 we have d j = 2 k−1 if and only if (…”