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PreprintCC BY 4.0arXiv

The strong Ramanujan bound for Drinfeld modular forms

  • Sjoerd de Vries
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Abstract

The Ramanujan bound states that the degree of the trace of the p-Hecke operator on a space of weight k Drinfeld cusp forms is at most d(k−2)/2, where p is a prime of degree d in 𝔽q[T]. We prove the strong Ramanujan bound, which states that at level 1, this bound can be improved to d(k−(q+1))/2. In addition, we classify when the strong Ramanujan bound is sharp; surprisingly, this turns out to depend on p only through the parity of its degree. We also prove results related to the smallest slope of the Atkin operator and mod p distributions of Frobenius polynomials of Drinfeld modules over finite fields.

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Cite

BibTeXRIS
BibTeX
@misc{vries2026strong,
  title = {{The strong Ramanujan bound for Drinfeld modular forms}},
  author = {Sjoerd de Vries},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.14458},
  eprint = {2610.14458},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.14458},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - The strong Ramanujan bound for Drinfeld modular forms
AU  - Sjoerd de Vries
PY  - 2026
DA  - 2026/10/10
PB  - arXiv
DO  - 10.48550/arxiv.2610.14458
UR  - https://arxiv.org/abs/2610.14458
AB  - The Ramanujan bound states that the degree of the trace of the p-Hecke operator on a space of weight k Drinfeld cusp forms is at most d(k−2)/2, where p is a prime of degree d in 𝔽_q[T]. We prove the strong Ramanujan bound, which states that at level 1, this bound can be improved to d(k−(q+1))/2. In addition, we classify when the strong Ramanujan bound is sharp; surprisingly, this turns out to depend on p only through the parity of its degree. We also prove results related to the smallest slope of the Atkin operator and mod p distributions of Frobenius polynomials of Drinfeld modules over finite fields.
ER  -