Rank-Two Drinfeld Modules over Elliptic Coordinate Rings

Abstract

Let A be the affine coordinate ring of E∖{∞}, where E is an elliptic curve over 𝔽_q and ∞ is rational. We solve the rank-two case of the open problem of explicitly classifying sign-normalized Drinfeld A-modules in generic characteristic with a fixed determinant motive; the classification for rank greater than two remains open. Outside an explicit exceptional locus, the rank-two Drinfeld modules are parametrized by the open modular curve Y^q+1=π(X) with Y≠0. The polynomial π is squarefree, and the smooth projective completion of this curve has genus q². The roots of π split into two types. Above roots of the first type, all coefficient functions extend regularly, and their extensions yield exactly the exceptional modules. Roots of the second type correspond to cusps of the modular curve; above these roots, the coefficient family has poles.

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BibTeX RIS

BibTeX

@misc{hu2026ranktwo,
  title = {{Rank-Two Drinfeld Modules over Elliptic Coordinate Rings}},
  author = {Chuangqiang Hu and Ze-Dong Huang and Chang-An Zhao},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2606.26851},
  eprint = {2606.26851},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2606.26851},
}

RIS

TY  - UNPB
TI  - Rank-Two Drinfeld Modules over Elliptic Coordinate Rings
AU  - Chuangqiang Hu
AU  - Ze-Dong Huang
AU  - Chang-An Zhao
PY  - 2026
DA  - 2026/10/28
PB  - arXiv
DO  - 10.48550/arxiv.2606.26851
UR  - https://arxiv.org/abs/2606.26851
AB  - Let A be the affine coordinate ring of E∖{∞}, where E is an elliptic curve over 𝔽_q and ∞ is rational. We solve the rank-two case of the open problem of explicitly classifying sign-normalized Drinfeld A-modules in generic characteristic with a fixed determinant motive; the classification for rank greater than two remains open. Outside an explicit exceptional locus, the rank-two Drinfeld modules are parametrized by the open modular curve Y^q+1=π(X) with Y≠0. The polynomial π is squarefree, and the smooth projective completion of this curve has genus q^2. The roots of π split into two types. Above roots of the first type, all coefficient functions extend regularly, and their extensions yield exactly the exceptional modules. Roots of the second type correspond to cusps of the modular curve; above these roots, the coefficient family has poles.
ER  -

Abstract

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