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PreprintarXiv licencearXiv

Towards a twistless converse theorem

  • Andrew R. Booker
  • Min Lee
  • Nina Zubrilina
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Abstract

We prove a finiteness theorem for Dirichlet series satisfying the Euler product, analytic continuation, and functional equation expected of a holomorphic newform. For any fixed level N, even weight k, and exponent σ, only finitely many such series have Dirichlet coefficients ≪ nσ. No hypotheses on twists by Dirichlet characters are assumed. The proof combines a combinatorial spanning theorem for Ramanujan sums with an operator-theoretic argument that uses two distinct primes to establish discreteness of the possible Hecke eigenvalues.

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Cite

BibTeXRIS
BibTeX
@misc{booker2026towards,
  title = {{Towards a twistless converse theorem}},
  author = {Andrew R. Booker and Min Lee and Nina Zubrilina},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.07921},
  eprint = {2611.07921},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.07921},
}
RIS
TY  - UNPB
TI  - Towards a twistless converse theorem
AU  - Andrew R. Booker
AU  - Min Lee
AU  - Nina Zubrilina
PY  - 2026
DA  - 2026/11/05
PB  - arXiv
DO  - 10.48550/arxiv.2611.07921
UR  - https://arxiv.org/abs/2611.07921
AB  - We prove a finiteness theorem for Dirichlet series satisfying the Euler product, analytic continuation, and functional equation expected of a holomorphic newform. For any fixed level N, even weight k, and exponent σ, only finitely many such series have Dirichlet coefficients ≪ n^σ. No hypotheses on twists by Dirichlet characters are assumed. The proof combines a combinatorial spanning theorem for Ramanujan sums with an operator-theoretic argument that uses two distinct primes to establish discreteness of the possible Hecke eigenvalues.
ER  -