Research article

Computing mod $ \ell $ Galois representations associated to modular forms for small primes

  • Received: 25 June 2023 Revised: 22 September 2023 Accepted: 10 October 2023 Published: 23 October 2023
  • MSC : 11F80, 11F33, 11G30, 11G18

  • In this paper, we propose an algorithm for computing mod $ \ell $ Galois representations associated to modular forms of weight $ k $ when $ \ell < k-1 $. We also present the corresponding results for the projective Galois representations. Moreover, we apply our algorithms to explicitly compute the mod $ \ell $ projective Galois representations associated to $ \Delta_{k} $ for $ k = 16, 20, 22, 26 $ and all the unexceptional primes $ \ell $, with $ \ell < k-1 $. As an application, for $ k = 16, 20, 22, 26 $, we obtain the new bounds $ B_k $ of $ n $ such that $ a_n(\Delta_k)\ne0 $ for all $ n < B_k $.

    Citation: Peng Tian, Ha Thanh Nguyen Tran, Dung Hoang Duong. Computing mod $ \ell $ Galois representations associated to modular forms for small primes[J]. AIMS Mathematics, 2023, 8(12): 28766-28779. doi: 10.3934/math.20231473

    Related Papers:

  • In this paper, we propose an algorithm for computing mod $ \ell $ Galois representations associated to modular forms of weight $ k $ when $ \ell < k-1 $. We also present the corresponding results for the projective Galois representations. Moreover, we apply our algorithms to explicitly compute the mod $ \ell $ projective Galois representations associated to $ \Delta_{k} $ for $ k = 16, 20, 22, 26 $ and all the unexceptional primes $ \ell $, with $ \ell < k-1 $. As an application, for $ k = 16, 20, 22, 26 $, we obtain the new bounds $ B_k $ of $ n $ such that $ a_n(\Delta_k)\ne0 $ for all $ n < B_k $.



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    [1] S. J. Edixhoven, J. M. Couveignes, R. S. de Jong, F. Merkl, J. G. Bosman, Computational Aspects of Modular Forms and Galois Representations, Ann. of Math. Stud., 176, Princeton Univ. Press, Princeton, 2011.
    [2] P. Bruin, Modular curves, Arakelov theory, algorithmic applications, Ph.D. thesis, Universiteit Leiden, 2008.
    [3] N. Mascot, Computing modular Galois representations, Rendiconti del Circolo Matematico di Palermo, 62 (2013), 451–476. https://doi.org/10.1007/s12215-013-0136-4 doi: 10.1007/s12215-013-0136-4
    [4] P. Tian, Computations of Galois representations associated to modular forms of level one, Acta Arith., 164 (2014), 399–412. https://doi.org/10.4064/aa164-4-5 doi: 10.4064/aa164-4-5
    [5] M. Derickx, M. van Hoeij, J. Zeng, Computing Galois representations and equations for modular curves $X_H(\ell)$, http://arXiv.org/abs/1312.6819
    [6] J. Sturm, On the congruence of modular forms, Lect. Notes Math., 1240 (1987), 275–280. https://doi.org/10.1007/BFb0072985 doi: 10.1007/BFb0072985
    [7] D. H. Lehmer, The vanishing of Ramanujan's function $\tau(n)$, Duke Math. J., 10 (1947), 429–433. https://doi.org/10.1215/S0012-7094-47-01436-1 doi: 10.1215/S0012-7094-47-01436-1
    [8] J. P. Serre, Une interpr$\acute{e}$tation des congruences relatives $\grave{a}$ la fonction de Ramanujan, S$\acute{e}$minaire Delange-Pisot-Poitiou, 14, 1968.
    [9] P. Tian, H. Qin, Non-vanishing Fourier coefficients of $\Delta_{k}$, Appl. Math. Computat., 339 (2018), 507–515. https://doi.org/10.1016/j.amc.2018.07.022 doi: 10.1016/j.amc.2018.07.022
    [10] SAGE, Open source mathematics software, http://sagemath.org
    [11] J. P. Serre, Formes modulaires et fonctions z$\hat{e}$ta p-adiques, Lect. Notes Math., 350 (1973), 191–268. https://doi.org/10.1007/978-3-540-37802-0_4 doi: 10.1007/978-3-540-37802-0_4
    [12] H. P. F. Swinnerton-Dyer, On $\ell$-adic representations and congruences for coefficients of modular forms (I), Lect. Notes Math., 350 (1973), 1–55. https://doi.org/10.1007/BFb0072985 doi: 10.1007/BFb0072985
    [13] N. M. Katz, $p$-adic properties of modular schemes and modular forms, Lect. Notes Math., 350 (1973), 69–190. https://doi.org/10.1007/978-3-540-37802-0_3 doi: 10.1007/978-3-540-37802-0_3
    [14] B. H. Gross, A tameness criterion for Galois representations associated to modular forms (MOD $p$), Duke Math. J., 61 (1990), 445–517. https://doi.org/10.1215/S0012-7094-90-06119-8 doi: 10.1215/S0012-7094-90-06119-8
    [15] S. J. Edixhoven, The weight in Serre's conjectures on modular forms, Invent. Math., 109 (1992), 563–594. https://doi.org/10.1007/BF01232041 doi: 10.1007/BF01232041
    [16] N. M. Katz, A result on modular forms in characteristic $p$, Lect. Notes Math., 601 (1976), 53–61. https://doi.org/10.1007/BFb0063944 doi: 10.1007/BFb0063944
    [17] W. Kohnen, On Fourier coefficients of modular forms of different weights, Acta Arith., 113 (1971), 57–67.
    [18] P. Deligne, Formes modulaires et représentations $\ell$-adiques, Lect. Notes Math., 179 (1971), 139–172. https://doi.org/10.1007/BFb0058810 doi: 10.1007/BFb0058810
    [19] K. A. Ribet, Report on mod $\ell$ representations of $Gal(\bar{\mathbb{Q}}/\mathbb{Q})$, Motives (Seattle, WA, 1991), Amer. Math. Soc., Providence, RI, 1994,639–676.
    [20] K. A. Ribet, W. A. Stein, Lectures on Serre's conjectures, Arithmetic algebraic geometry (Park City, UT, 1999), Amer. Math. Soc., Providence, RI, 2001, 143–232.
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