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Preprints, Working Papers, ... Year : 2024

Using modular polynomials for eta products to compute isogenies

Abstract

Let E be an elliptic curve over a field K and a prime. There exists an elliptic curve E * related to E by an isogeny of degree only if Φ (X, j(E)) = 0, where Φ (X, Y) is the traditional modular polynomial. Moreover, this polynomial gives the coefficients of E * , together with parameters needed to build the isogeny explicitly. Since the traditional modular polynomial has large coefficients, many families with smaller coefficients can be used instead, as described by Elkies, Atkin and others. In this work, we concentrate on the computation of modular polynomials for eta products, as considered by Fricke. We review and complete the properties of these eta products to be able to perform the computations of these polynomials using classical algorithms. We give algebraic formulas à la Atkin to compute the coefficients of E *. Numerical examples and comparisons to other families are also given.
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hal-04423470 , version 1 (29-01-2024)

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  • HAL Id : hal-04423470 , version 1

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François Morain. Using modular polynomials for eta products to compute isogenies. 2024. ⟨hal-04423470⟩
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