Computing modular polynomials in quasi-linear time

We analyse and compare the complexity of several algorithms for computing modular polynomials. We show that an algorithm relying on floating point evaluation of modular functions and on interpolation, which has received little attention in the literature, has a complexity that is essentially (up to logarithmic factors) linear in the size of the computed polynomials. In particular, it obtains the classical modular polynomials Φl of prime level l in time O(l3 log4 l log log l). Besides treating modular polynomials for Γ(l), which are an important ingredient in many algorithms dealing with isogenies of elliptic curves, the algorithm is easily adapted to more general situations. Composite levels are handled just as easily as prime levels, as well as polynomials between a modular function and its transform of prime level, such as the Schläfli polynomials and their generalisations. Our distributed implementation of the algorithm confirms the theoretical analysis by computing modular equations of record level around 10000 in less than two weeks on ten processors. 1 Definitions and main result Modular polynomials, in their broadest sense, are bivariate polynomials with a pair of modular functions as zero. Given any two modular functions f and g for arbitrary congruence subgroups, the function fields C(f) and C(g) are finite extensions of C(j), so that there are two polynomials relating f resp. g to j. Taking the resultant of these polynomials with respect to j, one sees that a polynomial relationship between f and g exists. In practice, one is rather interested in the minimal polynomial of f over C(g), say, that will be called the modular polynomial of f with respect to g. If the functions satisfy conditions on the rationality and integrality of their q-expansion coefficients, Hasse’s principle ensures that the modular polynomial has rational integral coefficients. Different modular polynomials parameterise moduli spaces related to elliptic curves. Let Γ = Sl2(Z)/{±1} = PSl2(Z) be the full modular group, and CΓ = C(j) the field of modular functions invariant under Γ; j itself parameterises isomorphism classes of elliptic curves. Of special interest for applications is the congruence subgroup