Skip to content
DraconiticResearch
Search Sources API
182,460 studies31 sourcesUpdated 44 min ago
PreprintCC BY 4.0arXiv

On the Number of Hecke Eigenvalues of the Same Sign on GLn

  • Jesse Jääsaari
Open at sourcePDFDOICite

Abstract

We study the distribution of signs of the (real-valued) Hecke eigenvalues $A(m,1,...,1)$ of self-dual Hecke--Maass cusp forms for the group SLn(ℤ), where n≥ 2 is an integer. Our main result establishes, under the Generalised Ramanujan--Petersson Conjecture, that for all X≤ x≤ 2X outside a set of measure o(X) the short interval (x,x+H] contains at least H(log X)1/n2−1 indices with both positive and negative Hecke eigenvalues, provided that (log X)1−1/n2+ε≤ H≤ X/10. For GL2 Hecke--Maass cusp forms we obtain a stronger bound unconditionally, and for self-dual GL3 forms we obtain the same logarithmic exponent unconditionally. We also establish results under weaker bounds towards the Generalised Ramanujan--Petersson Conjecture together with the Generalised Lindelöf Hypothesis, and obtain lower bounds for the number of sign changes and for the number of Hecke eigenvalues of each sign in long intervals.

Related

  1. Hecke Eigenvalues of Ikeda Lifts
    Nagarjuna Chary Addanki, Ameya Pitale
    arXivPreprintCC BY 4.0Full textAlgebra & number theoryarXiv
  2. Square integrable representations and Hecke eigenvalues over a local non-archimedian field
    Alexander Braverman, David Kazhdan
    arXivPreprintCC BY 4.0Full textAlgebra & number theoryarXiv
  3. Sign-indefinite transmission conditions and near-zero eigenvalues and resonances
    Andrea Mantile, Andrea Posilicano
    arXivPreprintCC BY 4.0Full textAnalysis & PDEsMathematical physicsarXiv
  4. Non-vanishing mod p of derived Hecke algebras of multiplicative groups over number fields, II
    Jaesung Kwon
    arXivPreprintCC BY 4.0Full textAlgebra & number theoryarXiv
  5. The number of Laplacian eigenvalues of trees less than one
    Jiaxin Guo, Tao Hu, Quanyu Tang
    arXivPreprintCC BY 4.0Full textCombinatorics & discrete mathematicsarXiv

Cite

BibTeXRIS
BibTeX
@misc{jaasaari2026number,
  title = {{On the Number of Hecke Eigenvalues of the Same Sign on $\mathrmGL_n$}},
  author = {Jesse Jääsaari},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2609.10446},
  eprint = {2609.10446},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.10446},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - On the Number of Hecke Eigenvalues of the Same Sign on GL_n
AU  - Jesse Jääsaari
PY  - 2026
DA  - 2026/10/13
PB  - arXiv
DO  - 10.48550/arxiv.2609.10446
UR  - https://arxiv.org/abs/2609.10446
AB  - We study the distribution of signs of the (real-valued) Hecke eigenvalues $A(m,1,...,1)$ of self-dual Hecke--Maass cusp forms for the group SL_n(ℤ), where n≥ 2 is an integer. Our main result establishes, under the Generalised Ramanujan--Petersson Conjecture, that for all X≤ x≤ 2X outside a set of measure o(X) the short interval (x,x+H] contains at least H(log X)^1/n^2−1 indices with both positive and negative Hecke eigenvalues, provided that (log X)^1−1/n^2+ε≤ H≤ X/10. For GL_2 Hecke--Maass cusp forms we obtain a stronger bound unconditionally, and for self-dual GL_3 forms we obtain the same logarithmic exponent unconditionally. We also establish results under weaker bounds towards the Generalised Ramanujan--Petersson Conjecture together with the Generalised Lindelöf Hypothesis, and obtain lower bounds for the number of sign changes and for the number of Hecke eigenvalues of each sign in long intervals.
ER  -