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Computing modular Galois representations. (English) Zbl 1339.11065

Summary: 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 [B. Edixhoven (ed.) and J.-M. Couveignes (ed.) [Computational aspects of modular forms and Galois representations. How one can compute in polynomial time the value of Ramanujan’s tau at a prime. Princeton, NJ: Princeton University Press (2011; Zbl 1216.11004)]. 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 cusp forms 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 [J. Bosman, On the computation of Galois representations associated to level one modular forms.
url{arxiv: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.

MSC:

11F80 Galois representations
11F11 Holomorphic modular forms of integral weight
11F30 Fourier coefficients of automorphic forms
11G18 Arithmetic aspects of modular and Shimura varieties
11Y35 Analytic computations

Citations:

Zbl 1216.11004

Software:

SageMath; ecdata
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

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