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PreprintCC BY 4.0arXiv

Primitive Fourier Coefficients of Hermitian Klingen-Eisenstein Series

  • Nobuki Takeda
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Abstract

Let E/F be a CM extension and let 1≤ r<n. We determine primitive Fourier coefficients of Hermitian Klingen-Eisenstein series on Un,n induced from cusp forms on Ur,r, at principal congruence level. Using the pullback formula for Hermitian Eisenstein series, we express a normalized primitive Fourier coefficient of a Klingen-Eisenstein series as the corresponding primitive Fourier coefficient of a Siegel Eisenstein series multiplied by a Petersson inner product with an explicitly constructed holomorphic automorphic form. The automorphic form in the Petersson inner product is a finite sum of products of Hermitian theta series and Siegel Eisenstein series of degree r. Under some hypotheses, we also prove the p-integrality of these normalized primitive Fourier coefficients.

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Cite

BibTeXRIS
BibTeX
@misc{takeda2026primitive,
  title = {{Primitive Fourier Coefficients of Hermitian Klingen-Eisenstein Series}},
  author = {Nobuki Takeda},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.08370},
  eprint = {2610.08370},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.08370},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Primitive Fourier Coefficients of Hermitian Klingen-Eisenstein Series
AU  - Nobuki Takeda
PY  - 2026
DA  - 2026/10/06
PB  - arXiv
DO  - 10.48550/arxiv.2610.08370
UR  - https://arxiv.org/abs/2610.08370
AB  - Let E/F be a CM extension and let 1≤ r<n. We determine primitive Fourier coefficients of Hermitian Klingen-Eisenstein series on U_n,n induced from cusp forms on U_r,r, at principal congruence level. Using the pullback formula for Hermitian Eisenstein series, we express a normalized primitive Fourier coefficient of a Klingen-Eisenstein series as the corresponding primitive Fourier coefficient of a Siegel Eisenstein series multiplied by a Petersson inner product with an explicitly constructed holomorphic automorphic form. The automorphic form in the Petersson inner product is a finite sum of products of Hermitian theta series and Siegel Eisenstein series of degree r. Under some hypotheses, we also prove the p-integrality of these normalized primitive Fourier coefficients.
ER  -