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Uncountably many settled elements are never Frobenius elements
Abstract
Boston and Jones conjectured that given a number field K, a quadratic polynomial f∈ K[x], and an element α∈ K such that the arboreal Galois representation ρf,α:Gal(K̅/K)→ Aut(T∞) attached to the pair (f,α) is well defined, any element in the image of such representation that is a Frobenius for an unramified prime of K is a settled element. We exhibit an explicit uncountable subset S⊆ Aut(T∞), consisting of pairwise non-conjugate settled elements, such that no matter how K,f,α are chosen, no element of S can be a Frobenius for any ramified or unramified prime of K.
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BibTeX
@misc{ferraguti2026uncountably,
title = {{Uncountably many settled elements are never Frobenius elements}},
author = {Andrea Ferraguti},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.27997},
eprint = {2610.27997},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.27997},
}
RIS
TY - UNPB
TI - Uncountably many settled elements are never Frobenius elements
AU - Andrea Ferraguti
PY - 2026
DA - 2026/10/20
PB - arXiv
DO - 10.48550/arxiv.2610.27997
UR - https://arxiv.org/abs/2610.27997
AB - Boston and Jones conjectured that given a number field K, a quadratic polynomial f∈ K[x], and an element α∈ K such that the arboreal Galois representation ρ_f,α:Gal(K̅/K)→ Aut(T_∞) attached to the pair (f,α) is well defined, any element in the image of such representation that is a Frobenius for an unramified prime of K is a settled element. We exhibit an explicit uncountable subset S⊆ Aut(T_∞), consisting of pairwise non-conjugate settled elements, such that no matter how K,f,α are chosen, no element of S can be a Frobenius for any ramified or unramified prime of K.
ER -