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PreprintCC BY 4.0arXiv

Spectral L2-bounds on Eisenstein series and Diophantine properties of quadratic forms

  • Jitendra Bajpai
  • Kristian Holm
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Abstract

We give L2-bounds with respect to the height of the spectral parameter for Epstein zeta functions ZQ(s) and non-holomorphic Eisenstein series Ec(s,z) attached to the Hecke congruence subgroup Γ0 (p) of prime level p. Our bounds are completely explicit in the level and, additionally, constitute an almost sure logarithmic improvement over previously established bounds for the level-one Eisenstein series. Following the approach of Kelmer--Kontorovich--Lutsko, we combine the approximate functional equation for ZQ(s) with estimates for the number of pairs of lattice vectors at which Q takes nearly equal values. Under a Diophantine hypothesis on Q, we sharpen these counting estimates using results of Eskin--Margulis--Mozes on the distribution of values of quadratic forms of signature (2,2).

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Cite

BibTeXRIS
BibTeX
@misc{bajpai2026spectral,
  title = {{Spectral $L^2$-bounds on Eisenstein series and Diophantine properties of quadratic forms}},
  author = {Jitendra Bajpai and Kristian Holm},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.37420},
  eprint = {2610.37420},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.37420},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Spectral L^2-bounds on Eisenstein series and Diophantine properties of quadratic forms
AU  - Jitendra Bajpai
AU  - Kristian Holm
PY  - 2026
DA  - 2026/10/27
PB  - arXiv
DO  - 10.48550/arxiv.2610.37420
UR  - https://arxiv.org/abs/2610.37420
AB  - We give L^2-bounds with respect to the height of the spectral parameter for Epstein zeta functions Z_Q(s) and non-holomorphic Eisenstein series E_c(s,z) attached to the Hecke congruence subgroup Γ_0 (p) of prime level p. Our bounds are completely explicit in the level and, additionally, constitute an almost sure logarithmic improvement over previously established bounds for the level-one Eisenstein series. Following the approach of Kelmer--Kontorovich--Lutsko, we combine the approximate functional equation for Z_Q(s) with estimates for the number of pairs of lattice vectors at which Q takes nearly equal values. Under a Diophantine hypothesis on Q, we sharpen these counting estimates using results of Eskin--Margulis--Mozes on the distribution of values of quadratic forms of signature (2,2).
ER  -