Skip to Main content Skip to Navigation
Journal articles

Computing modular Galois representations

Nicolas Mascot 1, 2, *
* Corresponding author
1 LFANT - Lithe and fast algorithmic number theory
IMB - Institut de Mathématiques de Bordeaux, Inria Bordeaux - Sud-Ouest
Abstract : We compute modular Galois representations associated with a newform $f$, and study the related problem of computing the coefficients of $f$ modulo a small prime $\ell$. To this end, we design a practical variant of the complex approximations method presented in the book edited by B. Edixhoven and J.-M. Couveignes. Its efficiency stems from several new ingredients. For instance, we use fast exponentiation in the modular jacobian instead of analytic continuation, which greatly reduces the need to compute abelian integrals, since most of the computation handles divisors. Also, we introduce an efficient way to compute arithmetically well-behaved functions on jacobians, a method to expand cuspforms in quasi-linear time, and a trick making the computation of the image of a Frobenius element by a modular Galois representation more effective. We illustrate our method on the newforms $\Delta$ and $E_4 \Delta$, and manage to compute for the first time the associated faithful representations modulo $\ell$ and the values modulo $\ell$ of Ramanujan's $\tau$ function at huge primes for $\ell \in {11,13,17,19,29}$. In particular, we get rid of the sign ambiguity stemming from the use of a non-faithful representation as in J. Bosman's work. As a consequence, we can compute the values of $\tau(p) \bmod 2^11.3^6.5^3.7.11.13.17.19.23.29.691 \approx 2.8.10^19$ for huge primes $p$. These representations lie in the jacobian of modular curves of genus up to 22.
Complete list of metadata

Cited literature [27 references]  Display  Hide  Download

https://hal.archives-ouvertes.fr/hal-01110451
Contributor : Nicolas Mascot Connect in order to contact the contributor
Submitted on : Wednesday, January 28, 2015 - 11:23:32 AM
Last modification on : Thursday, January 20, 2022 - 5:31:36 PM
Long-term archiving on: : Saturday, April 15, 2017 - 9:38:19 PM

File

Article 3.pdf
Files produced by the author(s)

Identifiers

Collections

Citation

Nicolas Mascot. Computing modular Galois representations. Rendiconti del Circolo Matematico di Palermo, 2013, 62 (3), pp.451 - 476. ⟨10.1007/s12215-013-0136-4⟩. ⟨hal-01110451⟩

Share

Metrics

Les métriques sont temporairement indisponibles