Adelic Galois representations are maximal for almost all Drinfeld modules

Abstract

Let r≥ 3 be an integer, let q be an odd prime power, and set A=𝔽_q[T] and F=𝔽_q(T). We study the Galois representations associated with rank r Drinfeld modules of the form ϕ_T=T+g₁τ+⋯+g_rτ^r, where g_i∈ A and g_r≠ 0. Ordering the coefficient tuples g⃗=(g₁,…,g_r) by their maximum degree, we prove that, for a density-one set of tuples, the λ-adic Galois representation is surjective for every prime λ of A, and the adelic Galois image contains SL_r(Â). Moreover, for every tuple in this set, the index of the adelic image equals the index of the corresponding rank-one determinant representation and hence divides q−1. These results extend Zywina's theorem in rank $2$ to arbitrary ranks r≥ 3. As a further consequence, we establish the existence of rank r Drinfeld modules with surjective adelic Galois representations for every r≥ 3, extending previous existence results which were known for prime values of r.

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BibTeX RIS

BibTeX

@misc{ray2026adelic,
  title = {{Adelic Galois representations are maximal for almost all Drinfeld modules}},
  author = {Anwesh Ray},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.01464},
  eprint = {2611.01464},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.01464},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Adelic Galois representations are maximal for almost all Drinfeld modules
AU  - Anwesh Ray
PY  - 2026
DA  - 2026/11/01
PB  - arXiv
DO  - 10.48550/arxiv.2611.01464
UR  - https://arxiv.org/abs/2611.01464
AB  - Let r≥ 3 be an integer, let q be an odd prime power, and set A=𝔽_q[T] and F=𝔽_q(T). We study the Galois representations associated with rank r Drinfeld modules of the form ϕ_T=T+g_1τ+⋯+g_rτ^r, where g_i∈ A and g_r≠ 0. Ordering the coefficient tuples g⃗=(g_1,…,g_r) by their maximum degree, we prove that, for a density-one set of tuples, the λ-adic Galois representation is surjective for every prime λ of A, and the adelic Galois image contains SL_r(Â). Moreover, for every tuple in this set, the index of the adelic image equals the index of the corresponding rank-one determinant representation and hence divides q−1. These results extend Zywina's theorem in rank $2$ to arbitrary ranks r≥ 3. As a further consequence, we establish the existence of rank r Drinfeld modules with surjective adelic Galois representations for every r≥ 3, extending previous existence results which were known for prime values of r.
ER  -

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