Hecke structure of quaternionic modular forms mod p

Abstract

We relate the systems of Hecke eigenvalues arising from the (mod p) modular forms on the Shimura curve attached to the indefinite quaternion algebra B of discriminant δ over Q to the systems of Hecke eigenvalues arising from the (mod p) algebraic modular forms attached to the definite quaternion algebra D of discriminant pδ. Moreover, we discuss details of the Hecke structure on the latter spaces, following ideas of Serre. The entire setup can be seen as a Shimura curve analogue of Serre's letter to Tate on quaternions and modular forms. The bijection between the sets of systems of Hecke eigenvalues is a special case of recent work of Terakado and Yu; the novelty of this paper is the explicit nature of the construction, allowing for finer control of its behaviour with respect to weights, as well as the results on the Hecke module structure.

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BibTeX RIS

BibTeX

@misc{fam2026hecke,
  title = {{Hecke structure of quaternionic modular forms mod $p$}},
  author = {Yiannis Fam and Alexandru Ghitza},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.15495},
  eprint = {2609.15495},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.15495},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Hecke structure of quaternionic modular forms mod p
AU  - Yiannis Fam
AU  - Alexandru Ghitza
PY  - 2026
DA  - 2026/09/29
PB  - arXiv
DO  - 10.48550/arxiv.2609.15495
UR  - https://arxiv.org/abs/2609.15495
AB  - We relate the systems of Hecke eigenvalues arising from the (mod p) modular forms on the Shimura curve attached to the indefinite quaternion algebra B of discriminant δ over Q to the systems of Hecke eigenvalues arising from the (mod p) algebraic modular forms attached to the definite quaternion algebra D of discriminant pδ. Moreover, we discuss details of the Hecke structure on the latter spaces, following ideas of Serre. The entire setup can be seen as a Shimura curve analogue of Serre's letter to Tate on quaternions and modular forms. The bijection between the sets of systems of Hecke eigenvalues is a special case of recent work of Terakado and Yu; the novelty of this paper is the explicit nature of the construction, allowing for finer control of its behaviour with respect to weights, as well as the results on the Hecke module structure.
ER  -

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