Relations among Ramanujan-type congruences II: Ramanujan-type congruences in half-integral weights
-
Martin Raum
Abstract
We link Ramanujan-type congruences, which emerge abundantly in combinatorics, to the Galois- and geometric theory of modular forms. Specifically, we show that Ramanujan-type congruences are preserved by the action of the shallow Hecke algebra, and discover a dichotomy between congruences originating in Hecke eigenvalues and congruences on arithmetic progressions with cube-free periods. The latter provide congruences among algebraic parts of twisted central
Funding source: Vetenskapsrådet
Award Identifier / Grant number: 2015-04139
Award Identifier / Grant number: 2019-03551
Funding statement: The author was partially supported by Vetenskapsrådet Grants 2015-04139 and 2019-03551.
Acknowledgements
The author wishes to thank Scott Ahlgren, Olivia Beckwith, and Olav Richter for inspiring discussions and helpful comments. The author is also grateful to the referee for their helpful and exceptionally thorough report.
References
[1] S. Ahlgren, P. Allen and S. Tang, Congruences like Atkin’s for the partition function, Trans. Amer. Math. Soc. Ser. B 9 (2022), 1044–1064. 10.1090/btran/128Search in Google Scholar
[2] S. Ahlgren, O. Beckwith and M. Raum, Scarcity of congruences for the partition function, preprint (2020), https://arxiv.org/abs/2006.07645. Search in Google Scholar
[3] S. Ahlgren and K. Ono, Congruence properties for the partition function, Proc. Natl. Acad. Sci. USA 98 (2001), no. 23, 12882–12884. 10.1073/pnas.191488598Search in Google Scholar PubMed PubMed Central
[4]
A. O. L. Atkin,
Multiplicative congruence properties and density problems for
[5] J. H. Bruinier, Nonvanishing modulo l of Fourier coefficients of half-integral weight modular forms, Duke Math. J. 98 (1999), no. 3, 595–611. 10.1215/S0012-7094-99-09819-8Search in Google Scholar
[6] N. Dummigan, Lifting congruences to half-integral weight, Res. Number Theory 8 (2022), no. 3, Paper No. 59. 10.1007/s40993-022-00356-3Search in Google Scholar
[7] C. Fieker, W. Hart, T. Hofmann and F. Johansson, Oscar/Hecke (version 0.8.6), 2021, https://oscar.computeralgebra.de/. Search in Google Scholar
[8] F. Johansson, Efficient implementation of the Hardy–Ramanujan–Rademacher formula, LMS J. Comput. Math. 15 (2012), 341–359. 10.1112/S1461157012001088Search in Google Scholar
[9] L. J. P. Kilford, Generating spaces of modular forms with η-quotients, JP J. Algebra Number Theory Appl. 8 (2007), no. 2, 213–226. Search in Google Scholar
[10] W. Kohnen, Fourier coefficients of modular forms of half-integral weight, Math. Ann. 271 (1985), no. 2, 237–268. 10.1007/BF01455989Search in Google Scholar
[11] H. Maass, Lectures on Modular Functions of one Complex Variable, Tata Inst. Fundam. Res. Lectures on Math. 29, Tata Institute of Fundamental Research, Bombay, 1964. Search in Google Scholar
[12] Y. Maeda, A congruence between modular forms of half-integral weight, Hokkaido Math. J. 12 (1983), no. 1, 64–73. 10.14492/hokmj/1381757792Search in Google Scholar
[13] S. Niwa, Modular forms of half integral weight and the integral of certain theta-functions, Nagoya Math. J. 56 (1975), 147–161. 10.1017/S0027763000016445Search in Google Scholar
[14] K. Ono, Distribution of the partition function modulo m, Ann. of Math. (2) 151 (2000), no. 1, 293–307. 10.2307/121118Search in Google Scholar
[15] C.-S. Radu, A proof of Subbarao’s conjecture, J. Reine Angew. Math. 672 (2012), 161–175. 10.1515/CRELLE.2011.165Search in Google Scholar
[16] C.-S. Radu, Proof of a conjecture by Ahlgren and Ono on the non-existence of certain partition congruences, Trans. Amer. Math. Soc. 365 (2013), no. 9, 4881–4894. 10.1090/S0002-9947-2013-05777-7Search in Google Scholar
[17] S. Ramanujan, Congruence properties of partitions, Math. Z. 9 (1921), no. 1–2, 147–153. 10.1007/BF01378341Search in Google Scholar
[18] M. Raum, Relations among Ramanujan-type congruences I, Adv. Math. 409 (2022), Paper No. 108668. 10.1016/j.aim.2022.108668Search in Google Scholar
[19] The Sage Developers, SageMath, the Sage Mathematics Software System (Version 9.2), 2020, https://www.sagemath.org. Search in Google Scholar
[20] G. Shimura, On modular forms of half integral weight, Ann. of Math. (2) 97 (1973), 440–481. 10.2307/1970831Search in Google Scholar
[21] T. Shintani, On construction of holomorphic cusp forms of half integral weight, Nagoya Math. J. 58 (1975), 83–126. 10.1017/S0027763000016706Search in Google Scholar
[22] F. Strömberg, Weil representations associated with finite quadratic modules, Math. Z. 275 (2013), no. 1–2, 509–527. 10.1007/s00209-013-1145-xSearch in Google Scholar
[23] J.-L. Waldspurger, Sur les coefficients de Fourier des formes modulaires de poids demi-entier, J. Math. Pures Appl. (9) 60 (1981), no. 4, 375–484. Search in Google Scholar
[24] R. L. Weaver, New congruences for the partition function, Ramanujan J. 5 (2001), no. 1, 53–63. 10.1023/A:1011493128408Search in Google Scholar
[25]
Y. Yang,
Modular forms of half-integral weights on
© 2023 Walter de Gruyter GmbH, Berlin/Boston
Articles in the same Issue
- 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
Articles in the same Issue
- 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
