<i>p</i>-adic class invariants

Bröker, Reinier

doi:10.1112/s1461157009000175

Cited by 2 publications

(1 citation statement)

References 19 publications

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“…While A(z) and B(z) are not rational functions of j(z), we note that the curves defined by Ψ A (X, J) and Ψ B (X, J) both have genus 0 (and thus admit a rational parametrization, although we shall not make explicit use of this fact). It follows that, at least for discriminants prime to 6, the CM values of A(z) and B(z) are class invariants, and we can compute the class polynomials H D ( A; x) and H D (B; x) using standard algorithms such as those found in [5,14,16]. Under the GRH, we can compute these class polynomials in O(|D|) expected time, which is negligible compared to the O(|D| 5/2 ) expected running time of Algorithm 2.…”

Section: For Each Prime P ∈ Smentioning

confidence: 99%

Class polynomials for nonholomorphic modular functions

Bruinier

Ono

Sutherland

2016

Journal of Number Theory

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Abstract. 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 obtain CRT-based algorithms that compute the class polynomials H D (F ; x), whose roots are the discriminant D singular moduli for F (z). By applying these results to a specific weak Maass form F p (z), we obtain a CRT-based algorithm for computing partition class polynomials, a sequence of polynomials whose traces give the partition numbers p(n). Under the GRH, the expected running time of this algorithm is O(n 5/2+o(1) ). Key to these results is a fast CRTbased algorithm for computing the classical modular polynomial Φ m (X, Y ) that we obtain by extending the isogeny volcano approach previously developed for prime values of m.

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Section: For Each Prime P ∈ Smentioning

confidence: 99%

Class polynomials for nonholomorphic modular functions

Bruinier

Ono

Sutherland

2016

Journal of Number Theory

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On the evaluation of modular polynomials

Sutherland

2013

Open Book Series

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We present two algorithms that, given a prime`and an elliptic curve E=‫ކ‬ q , directly compute the polynomialˆ`.j.E/; Y / 2 ‫ކ‬ q OEY whose roots are the jinvariants of the elliptic curves that are`-isogenous to E. We do not assume that the modular polynomialˆ`.X; Y / is given. The algorithms may be adapted to handle other types of modular polynomials, and we consider applications to point counting and the computation of endomorphism rings. We demonstrate the practical efficiency of the algorithms by setting a new point-counting record, modulo a prime q with more than 5,000 decimal digits, and by evaluating a modular polynomial of level`D 100,019.

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p-adic class invariants

Abstract: We develop a new p-adic algorithm to compute the minimal polynomial of a class invariant. Our approach works for virtually any modular function yielding class invariants. The main algorithmic tool is modular polynomials, a concept which we generalize to functions of higher level.

Cited by 2 publications

References 19 publications

Class polynomials for nonholomorphic modular functions

Class polynomials for nonholomorphic modular functions

On the evaluation of modular polynomials

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