On the bounded-conductor finiteness conjecture in equal characteristic
Abstract
We investigate the equal-characteristic case of the Moon--Taguchi bounded-conductor finiteness conjecture for mod p Galois representations over global function fields. We first establish a conditional finiteness theorem: for any global function field K of characteristic p and any integer n ≥ 1, there are only finitely many isomorphism classes of continuous, semisimple, everywhere unramified, and geometric representations ρ: GK → GLn(F̅p) that admit an everywhere unramified characteristic-zero lift. Furthermore, we prove that the conjecture fails in general without this lifting hypothesis. Concretely, we construct infinitely many global function fields K of characteristic p admitting a continuous, surjective, absolutely irreducible, everywhere unramified, and geometric representation ρr: GK ↠ SL2(Fpr) for each integer r ≥ 4. As a corollary, we construct an everywhere unramified Galois extension L/K, regular over Fp, with Galois group Gal(L/K)≅ ∏r≥4PSL2(Fpr). Combined with known cross-characteristic finiteness theorems, this completely resolves the question posed by Moon and Taguchi in dimension two: such extensions exist over global function fields of characteristic p, whereas they cannot exist over global function fields of characteristic different from p.
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BibTeX
@misc{luo2026boundedconductor,
title = {{On the bounded-conductor finiteness conjecture in equal characteristic}},
author = {Yufan Luo and Yiqi Xu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.11456},
eprint = {2609.11456},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.11456},
note = {CC BY-SA 4.0},
}
RIS
TY - UNPB
TI - On the bounded-conductor finiteness conjecture in equal characteristic
AU - Yufan Luo
AU - Yiqi Xu
PY - 2026
DA - 2026/09/10
PB - arXiv
DO - 10.48550/arxiv.2609.11456
UR - https://arxiv.org/abs/2609.11456
AB - We investigate the equal-characteristic case of the Moon--Taguchi bounded-conductor finiteness conjecture for mod p Galois representations over global function fields. We first establish a conditional finiteness theorem: for any global function field K of characteristic p and any integer n ≥ 1, there are only finitely many isomorphism classes of continuous, semisimple, everywhere unramified, and geometric representations ρ: G_K → GL_n(F̅_p) that admit an everywhere unramified characteristic-zero lift. Furthermore, we prove that the conjecture fails in general without this lifting hypothesis. Concretely, we construct infinitely many global function fields K of characteristic p admitting a continuous, surjective, absolutely irreducible, everywhere unramified, and geometric representation ρ_r: G_K ↠ SL_2(F_p^r) for each integer r ≥ 4. As a corollary, we construct an everywhere unramified Galois extension L/K, regular over F_p, with Galois group Gal(L/K)≅ ∏_r≥4PSL_2(F_p^r). Combined with known cross-characteristic finiteness theorems, this completely resolves the question posed by Moon and Taguchi in dimension two: such extensions exist over global function fields of characteristic p, whereas they cannot exist over global function fields of characteristic different from p.
ER -