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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 $Phi_ell$ of prime level $ell$ in time O (ell^3 log^4 ell log log ell). Besides treating modular polynomials for $Gamma^0 (ell)$, 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 Schlafli 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.
We discuss practical and some theoretical aspects of computing a database of classical modular forms in the L-functions and Modular Forms Database (LMFDB).
We give algorithms for computing the singular moduli of suitable nonholomorphic modular functions F(z). By combining the theory of isogeny volcanoes with a beautiful observation of Masser concerning the nonholomorphic Eisenstein series E_2*(z), we ob
Let f in Q[z] be a polynomial of degree d at least two. The associated canonical height hat{h}_f is a certain real-valued function on Q that returns zero precisely at preperiodic rational points of f. Morton and Silverman conjectured in 1994 that the
We present a deterministic algorithm which computes the multilinear factors of multivariate lacunary polynomials over number fields. Its complexity is polynomial in $ell^n$ where $ell$ is the lacunary size of the input polynomial and $n$ its number o
We develop a new algorithm to compute a basis for $M_k(Gamma_0(N))$, the space of weight $k$ holomorphic modular forms on $Gamma_0(N)$, in the case when the graded algebra of modular forms over $Gamma_0(N)$ is generated at weight two. Our tests show