Computing modular Galois representations

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 \(22\).

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Notes

  1. 1.

    If we used a basis of eigenforms, the common number field containing the Fourier coefficients of all these forms could be much larger.

  2. 2.

    Here, the method breaks down for \(\ell = 13\). Indeed, this is the only case in which \(g_0 = 0\) (remember we supposed \(\ell \geqslant 11\)), so that there is no such form in this case. So, in this special case \(\ell = 13\), classical methods to expand the forms should be used instead. This is not a big problem, as this is a “small” case (\(g\) is only \(2\)), so little accuracy is needed and the whole Galois representation computation is quite fast anyway.

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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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Mascot, N. Computing modular Galois representations. Rend. Circ. Mat. Palermo 62, 451–476 (2013). https://doi.org/10.1007/s12215-013-0136-4

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Keywords

  • Galois representations
  • Modular forms
  • Algorithms
  • Complex approximations
  • Modular curves
  • Jacobian varieties

Mathematics Subject Classification (2010)

  • 11F80
  • 20C20
  • 11F11
  • 11F30
  • 11G18