Abstract
Recently, Chen and Kiming studied the theta operator on modular forms modulo prime powers
Funding source: National Research Foundation of Korea
Award Identifier / Grant number: 2019R1A6A1A11051177
Award Identifier / Grant number: 2020R1I1A1A01074746
Award Identifier / Grant number: NRF-2022R1A2C1003203
Funding statement: The authors were supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (grant no. 2019R1A6A1A11051177). In addition, Jigu Kim was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (grant no. 2020R1I1A1A01074746), and Yoonjin Lee was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MEST) (NRF-2022R1A2C1003203).
Acknowledgements
The authors deeply thank the reviewers for constructive comments and suggestions, which was helpful for improving the quality and readability of this paper.
References
[1] I. Chen and I. Kiming, On the theta operator for modular forms modulo prime powers, Mathematika 62 (2016), no. 2, 321–336. 10.1112/S0025579315000212Suche in Google Scholar
[2] I. Chen, I. Kiming and G. Wiese, On modular Galois representations modulo prime powers, Int. J. Number Theory 9 (2013), no. 1, 91–113. 10.1142/S1793042112501254Suche in Google Scholar
[3] S. R. Dahmen and S. Yazdani, Level lowering modulo prime powers and twisted Fermat equations, Canad. J. Math. 64 (2012), no. 2, 282–300. 10.4153/CJM-2011-059-8Suche in Google Scholar
[4] P. Deligne and M. Rapoport, Les schémas de modules de courbes elliptiques, Modular Functions of one Variable. II (Antwerp 1972), Lecture Notes in Math. 349, Springer, Berlin (1973), 143–316. 10.1007/978-3-540-37855-6_4Suche in Google Scholar
[5]
M. Dewar,
The image and kernel of Atkin’s
[6] F. Diamond and J. Im, Modular forms and modular curves, Seminar on Fermat’s Last Theorem (Toronto 1993–1994), CMS Conf. Proc. 17, American Mathematical Society, Providence (1995), 39–133. Suche in Google Scholar
[7] B. Edixhoven, The weight in Serre’s conjectures on modular forms, Invent. Math. 109 (1992), no. 3, 563–594. 10.1007/BF01232041Suche in Google Scholar
[8] B. H. Gross, A tameness criterion for Galois representations associated to modular forms (mod p), Duke Math. J. 61 (1990), no. 2, 445–517. 10.1215/S0012-7094-90-06119-8Suche in Google Scholar
[9] N. Jochnowitz, A study of the local components of the Hecke algebra mod l, Trans. Amer. Math. Soc. 270 (1982), no. 1, 253–267. 10.1090/S0002-9947-1982-0642340-0Suche in Google Scholar
[10] N. Jochnowitz, Congruences between systems of eigenvalues of modular forms, Trans. Amer. Math. Soc. 270 (1982), no. 1, 269–285. 10.1090/S0002-9947-1982-0642341-2Suche in Google Scholar
[11] N. M. Katz, p-adic properties of modular schemes and modular forms, Modular Functions of one Variable. III (Antwerp 1972), Lecture Notes in Math. 350, Springer, Berlin (1973), 69–190. 10.1007/978-3-540-37802-0_3Suche in Google Scholar
[12] N. M. Katz, A result on modular forms in characteristic p, Modular Functions of one Variable. V (Bonn 1976), Lecture Notes in Math. 601, Springer, Berlin, (1977), 53–61. 10.1007/BFb0063944Suche in Google Scholar
[13] C. Khare and J.-P. Wintenberger, Serre’s modularity conjecture. I, Invent. Math. 178 (2009), no. 3, 485–504. 10.1007/s00222-009-0205-7Suche in Google Scholar
[14] C. Khare and J.-P. Wintenberger, Serre’s modularity conjecture. II, Invent. Math. 178 (2009), no. 3, 505–586. 10.1007/s00222-009-0206-6Suche in Google Scholar
[15] I. Kiming, N. Rustom and G. Wiese, On certain finiteness questions in the arithmetic of modular forms, J. Lond. Math. Soc. (2) 94 (2016), no. 2, 479–502. 10.1112/jlms/jdw045Suche in Google Scholar
[16] C. Queen, The existence of p-adic Abelian L-functions, Number Theory and Algebra, Academic Press, New York (1977), 263–288. Suche in Google Scholar
[17] N. Rustom, Filtrations of dc-weak eigenforms, Acta Arith. 180 (2017), no. 4, 297–318. 10.4064/aa8491-8-2017Suche in Google Scholar
[18] J.-P. Serre, Formes modulaires et fonctions zêta p-adiques, Modular Functions of One Variable. III (Antwerp 1972), Lecture Notes in Math. 350, Springer, Berlin (1973), 191–268. 10.1007/978-3-540-37802-0_4Suche in Google Scholar
[19] J.-P. Serre, Valeurs propres des opérateurs de Hecke modulo l, Journées Arithmétiques de Bordeaux, Astérisque 24–25, Société Mathématique de France, Paris (1975), 109–117. Suche in Google Scholar
[20]
J.-P. Serre,
Sur les représentations modulaires de degré 2 de
[21] H. P. F. Swinnerton-Dyer, On l-adic representations and congruences for coefficients of modular forms, Modular Functions of one Variable. III (Antwerp 1972), Lecture Notes in Math. 350, Springer, Berlin (1973), 1–55. 10.1007/978-3-540-37802-0_1Suche in Google Scholar
[22] X. Taixés i Ventosa and G. Wiese, Computing congruences of modular forms and Galois representations modulo prime powers, Arithmetic, Geometry, Cryptography and Coding Theory 2009, Contemp. Math. 521, American Mathematical Society, Providence (2010), 145–166. 10.1090/conm/521/10279Suche in Google Scholar
[23] D. Zagier, Elliptic modular forms and their applications, The 1-2-3 of Modular Forms, Universitext, Springer, Berlin (2008), 1–103. 10.1007/978-3-540-74119-0_1Suche in Google Scholar
© 2023 Walter de Gruyter GmbH, Berlin/Boston
Artikel in diesem Heft
- Frontmatter
- The theta cycles for modular forms modulo prime powers
- Relations among Ramanujan-type congruences II: Ramanujan-type congruences in half-integral weights
- A unified approach to Gelfand and de Vries dualities
- The cardinality of orthogonal exponentials of planar self-affine measures with two-element digit set
- Categorically closed countable semigroups
- Symmetric skew braces and brace systems
- A new class of uncertainty principles for the k-Hankel wavelet transform
- Homogeneous ACM bundles on isotropic Grassmannians
- On the global L 2-boundedness of Fourier integral operators with rough amplitude and phase functions
- A local converse theorem for Archimedean GL(n)
- Fractional Bloom boundedness and compactness of commutators
- A Mikhlin-type multiplier theorem for the partial harmonic oscillator
- Rational pullbacks of toric foliations
- On fractional integrals generated by Radon transforms over paraboloids
Artikel in diesem Heft
- Frontmatter
- The theta cycles for modular forms modulo prime powers
- Relations among Ramanujan-type congruences II: Ramanujan-type congruences in half-integral weights
- A unified approach to Gelfand and de Vries dualities
- The cardinality of orthogonal exponentials of planar self-affine measures with two-element digit set
- Categorically closed countable semigroups
- Symmetric skew braces and brace systems
- A new class of uncertainty principles for the k-Hankel wavelet transform
- Homogeneous ACM bundles on isotropic Grassmannians
- On the global L 2-boundedness of Fourier integral operators with rough amplitude and phase functions
- A local converse theorem for Archimedean GL(n)
- Fractional Bloom boundedness and compactness of commutators
- A Mikhlin-type multiplier theorem for the partial harmonic oscillator
- Rational pullbacks of toric foliations
- On fractional integrals generated by Radon transforms over paraboloids