Abstract
In this paper, we investigate sign changes of Fourier coefficients of half-integral weight cusp forms. In a fixed square class \(t\mathbb {Z}^2\), we investigate the sign changes in the \(tp^2\)-th coefficient as p runs through the split or inert primes over the ring of integers in a quadratic extension of the rationals. We show that infinitely many sign changes occur in both sets of primes when there exists a prime dividing the discriminant of the field which does not divide the level of the cusp form and find an explicit condition that determines whether sign changes occur when every prime dividing the discriminant also divides the level.
This is a preview of subscription content, access via your institution.
References
- 1.
Apostol, T.: Introduction to Analytic Number Theory: 3rd Printing. Springer, New York (1986)
- 2.
Atkin, A., Lehner, J.: Hecke operators on \(\Gamma _0(m)\). Math. Ann. 185, 134–160 (1970)
- 3.
Bruinier, J., Kohnen, W.: Sign changes of coefficients of half integral weight modular forms in: Modular forms on Schiermonnikoong (eds. B. Edixhoven et. al.), 57–66, Cambridge Univ. Press, (2008)
- 4.
Deligne, P.: La conjecture de Weil I. Publ. Math. Inst. Hautes Études Sci. 43, 273–307 (1974)
- 5.
Duke, W., Schulze-Pillot, R.: Representation of integers by positive ternary quadratic forms and equidistribution of lattice points on ellipsoids. Invent. Math. 99, 49–57 (1990)
- 6.
Hanke, J.: Some recent results about (ternary) quadratic forms, in: Number Theory, in: CRM Proc. Lecture Notes, vol. 36, Amer. Math. Soc., Providence, 147–164 (2004)
- 7.
Iwaniec, H.: Topics in Classical Automorphic Forms, Graduate Studies in Mathematics 17. Amer. Math. Soc, Providence (1997)
- 8.
Iwaniec, H., Kowalski, E.: Analytic number theory, Amer. Math. Soc. Colloq. Publ. 53 Amer. Math. Soc., Providence, RI, (2004)
- 9.
Iwaniec, H., Kohnen, W., Sengupta, J.: The first sign change of Hecke eigenvalue. Int. J. Number Theory 3, 355–363 (2007)
- 10.
Kitaoka, Y.: Arithmetic of Quadratic Forms. Cambridge University Press, Cambridge (1993)
- 11.
Knopp, M., Kohnen, W., Pribitkin, W.: On the signs of Fourier coefficients of cusp forms. Rankin memorial issues. Ramanujan J. 7, 269–277 (2003)
- 12.
Koblitz, N.: Introduction to Elliptic Curves and Modular Forms. Springer, New York (1993)
- 13.
Kohnen, W., Sengupta, J.: On the first sign change of Hecke eigenvalues of newforms. Math. Z. 254, 173–184 (2006)
- 14.
Kohnen, W., Lau, Y.-K., Wu, J.: Fourier coefficients of cusp forms of half-integral weight. Math. Z. 273, 29–41 (2013)
- 15.
Lau, Y.-K., Wu, J.: The number of Hecke eigenvalues of same signs. Math. Z. 263, 957–970 (2009)
- 16.
Schulze-Pillot, R.: Representation by integral quadratic forms–a survey, Algebraic and arithmetic theory of quadratic forms, 303–321, Contemp. Math., 344, Amer. Math. Soc., Providence, RI, (2004)
- 17.
Shimura, G.: On modular forms of half integral weight. Ann. Math. (2) 97, 440–481 (1973)
Acknowledgements
The authors thank Yuk-Kam Lau for many helpful discussions and the anonymous referees for many useful corrections and comments.
Author information
Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
The research of the second author was supported by grants from the Research Grants Council of the Hong Kong SAR, China (Project Numbers HKU 17302515, 17316416, 17301317 and 17303618.)
Rights and permissions
About this article
Cite this article
He, Z., Kane, B. Sign Changes of Fourier Coefficients of Cusp Forms of Half-Integral Weight Over Split and Inert Primes in Quadratic Number Fields. Res. number theory 7, 10 (2021). https://doi.org/10.1007/s40993-020-00235-9
Received:
Accepted:
Published:
Keywords
- Half-integral weight modular forms
- Sign changes
- Fourier coefficients
- Quadratic number fields
- Quadratic forms
Mathematics Subject Classification
- 11F37
- 11F30
- 11N69
- 11R11
- 11E20