Sign Changes of Fourier Coefficients of Cusp Forms of Half-Integral Weight Over Split and Inert Primes in Quadratic Number Fields

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.

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Acknowledgements

The authors thank Yuk-Kam Lau for many helpful discussions and the anonymous referees for many useful corrections and comments.

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Correspondence to Ben Kane.

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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.)

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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

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Keywords

  • Half-integral weight modular forms
  • Sign changes
  • Fourier coefficients
  • Quadratic number fields
  • Quadratic forms

Mathematics Subject Classification

  • 11F37
  • 11F30
  • 11N69
  • 11R11
  • 11E20