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Drinfeld modular forms of higher rank from a lattice-oriented point of view

  • Liam Baker
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Abstract

A space ℒNr of Drinfeld modules of rank r ≥ 1 with level structure, or equivalently lattices of rank r with level structure, is introduced, and its irreducible components and group actions on it are investigated. A metric is defined on this space, its completion overleftarrowℒNr is established and the aforementioned group actions are extended to the completion. A decomposition of the completion into multiple smaller spaces ℒNs is proven. Drinfeld modular forms are defined as homogeneous holomorphic functions on ℒNr which are continuous on the completion overleftarrowℒNr, and the group actions above are extended to actions on the spaces of modular forms. Finally, the modular forms defined here are compared with those of Basson, Breuer, and Pink and with those of Gekeler, and it is shown that the cusp forms (those which are zero on the boundary) coincide.

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Cite

BibTeXRIS
BibTeX
@misc{baker2026drinfeld,
  title = {{Drinfeld modular forms of higher rank from a lattice-oriented point of view}},
  author = {Liam Baker},
  howpublished = {arXiv},
  year = {2026},
  month = aug,
  doi = {10.48550/arxiv.2608.28014},
  eprint = {2608.28014},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.28014},
}
RIS
TY  - UNPB
TI  - Drinfeld modular forms of higher rank from a lattice-oriented point of view
AU  - Liam Baker
PY  - 2026
DA  - 2026/08/28
PB  - arXiv
DO  - 10.48550/arxiv.2608.28014
UR  - https://arxiv.org/abs/2608.28014
AB  - A space ℒ_N^r of Drinfeld modules of rank r ≥ 1 with level structure, or equivalently lattices of rank r with level structure, is introduced, and its irreducible components and group actions on it are investigated. A metric is defined on this space, its completion overleftarrowℒ_N^r is established and the aforementioned group actions are extended to the completion. A decomposition of the completion into multiple smaller spaces ℒ_N^s is proven. Drinfeld modular forms are defined as homogeneous holomorphic functions on ℒ_N^r which are continuous on the completion overleftarrowℒ_N^r, and the group actions above are extended to actions on the spaces of modular forms. Finally, the modular forms defined here are compared with those of Basson, Breuer, and Pink and with those of Gekeler, and it is shown that the cusp forms (those which are zero on the boundary) coincide.
ER  -