On the computation of coefficients of modular forms: The reduction modulo $p$ approach

Zeng, Jinxiang; Yin, Li

doi:10.1090/s0025-5718-2014-02892-5

Cited by 8 publications

(16 citation statements)

References 27 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In [9, Corollar 7.4], Bosman's bound is 22798241520242687999 and our bound improves his by a factor approximately equal to 43. In the paper [22], Zeng also obtained the same prime. on this paper.…”

Section: Proofmentioning

confidence: 83%

See 1 more Smart Citation

Computations of Galois representations associated to modular forms of level one

Tian

2014

Acta Arith.

View full text Add to dashboard Cite

We propose an improved algorithm for computing mod ℓ Galois representations associated to a cusp form f of level one. The proposed method allows us to explicitly compute the case with ℓ = 29 and f of weight k = 16, and the cases with ℓ = 31 and f of weight k = 12, 20, 22. All the results are rigorously proved to be correct.As an example, we will compute the values modulo 31 of Ramanujan's tau function at some huge primes up to a sign. Also we will give an improved higher bound on Lehmer's conjecture for Ramanujan's tau function.

show abstract

Section: Proofmentioning

confidence: 83%

“…They took about 10 days for each of the cases with ℓ = 31 and one week for the case ℓ = 29. The polynomial P∆ 12 ,31 has also been obtained by Zeng [22]. His method avoids the high precision computations and is based on p-adic computations.…”

Section: Examplesmentioning

confidence: 99%

Computations of Galois representations associated to modular forms of level one

Tian

2014

Acta Arith.

View full text Add to dashboard Cite

show abstract

“…The classical approach, used in [CEC3], [EdiC14], [Bos07] and [Zen12], consists in selecting a rational function ξ on X 1 (ℓ) defined over Q and extending it to J 1 (ℓ) by Ξ :…”

Section: Evaluating the Torsion Divisorsmentioning

confidence: 99%

Computing modular Galois representations

Mascot

2013

Rend. Circ. Mat. Palermo

View full text Add to dashboard Cite

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 ℓ. To this end, we design a practical variant of the complex approximations method presented in [EC11]. 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 ∆ and E 4 · ∆, and manage to compute for the first time the associated faithful representations modulo ℓ and the values modulo ℓ of Ramanujan's τ function at huge primes for ℓ ∈ {11, 13, 17, 19, 29}. In particular, we get rid of the sign ambiguity stemming from the use of a non-faithful projective representation as in [Bos07]. As a consequence, we can compute the values of τ (p) mod 2 11 · 3 6 · 5 3 · 7 · 11 · 13 · 17 · 19 · 23 · 29 · 691 ≈ 2.8 · 10 19 for huge primes p. These representations lie in the jacobian of modular curves of genus up to 22.

show abstract

“…It is known [ZY15, Theorem 1.4] that when We also refer to [DvHZ13, Corollary 1.2], which claims that for all …”

Section: Introductionmentioning

confidence: 99%

Random ordering in modulus of consecutive Hecke eigenvalues of primitive forms

Bilu¹,

Deshouillérs²,

Gun³

et al. 2018

Compositio Math.

View full text Add to dashboard Cite

Let τ (·) be the classical Ramanujan τ -function and let k be a positive integer such that τ (n) = 0 for 1 ≤ n ≤ k/2. (This is known to be true for k < 10 23 , and, conjecturally, for all k.) Further, let σ be a permutation of the set {1, ..., k}. We show that there exist infinitely many positive integers m such that |τ (m + σ(1))| < |τ (m + σ(2))| < ... < |τ (m + σ(k))|.We also obtain a similar result for Hecke eigenvalues of primitive forms of square-free level.

show abstract

On the computation of coefficients of modular forms: The reduction modulo $p$ approach

Cited by 8 publications

References 27 publications

Computations of Galois representations associated to modular forms of level one

Computations of Galois representations associated to modular forms of level one

Computing modular Galois representations

Random ordering in modulus of consecutive Hecke eigenvalues of primitive forms

Contact Info

Product

Resources

About