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Rendiconti del Circolo Matematico di Palermo

, Volume 62, Issue 3, pp 451–476 | Cite as

Computing modular Galois representations

  • Nicolas MascotEmail author
Article

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 Edixhoven and Couveignes (Ann. of Math. Stud., vol. 176, Princeton University Press, Princeton, 2011). 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 \(E_4 \cdot \Delta \), 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 projective representation as in Bosman (On the computation of Galois representations associated to level one modular forms. arxiv.org/abs/0710.1237, 2007). As a consequence, we can compute the values of \(\tau (p)~\mathrm{mod}~2^{11} \times 3^6 \times 5^3 \times 7 \times 11 \times 13 \times 17 \times 19 \times 23 \times 29 \times 691 \approx 2.8 \times 10^{19}\) for huge primes \(p\). The representations we computed lie in the jacobian of modular curves of genus up to \(22\).

Keywords

Galois representations Modular forms Algorithms  Complex approximations Modular curves Jacobian varieties 

Mathematics Subject Classification (2010)

11F80 20C20 11F11 11F30 11G18 

Notes

Acknowledgments

I would like to heartily thank my advisor J.-M. Couveignes for offering me this beautiful subject to work on. More generally, I would like to thank people from the Bordeaux 1 university’s IMB for their support, with special thoughts to B. Allombert, K. Belabas, H. Cohen and A. Enge, as well as the PlaFRIM team. Finally, I thank B. Edixhoven for his remarks on earlier versions of this article, A. Page for helping me to make explicit the similarity classes in \(\mathrm{GL }_2({\mathbb {F}}_\ell )\), J. Klüners for his interest and assistance in formally proving that the polynomials I computed have the expected Galois group, and T. Selig for proofreading my English. This research was supported by the French ANR-12-BS01-0010-01 through the project PEACE, and by the DGA maîtrise de l’information. Experiments presented in this paper were carried out using the PlaFRIM experimental testbed, being developed under the Inria PlaFRIM development action with support from LABRI and IMB and other entities: Conseil Régional d’Aquitaine, FeDER, Université de Bordeaux and CNRS (see https://plafrim.bordeaux.inria.fr/).

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Copyright information

© Springer-Verlag Italia 2013

Authors and Affiliations

  1. 1.IMB, Université Bordeaux 1TalenceFrance
  2. 2.CNRS, IMB, UMR 5251TalenceFrance
  3. 3.INRIA, project LFANTTalenceFrance

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