A conjecture on Galois groups and large fields
Abstract
Let p be a prime number and let K be a field such that p is prime to the absolute Galois group Gal(K), in the sense that p does not divide the degree of any finite Galois extension of K. We conjecture that K is large in the sense of Pop: any smooth algebraic curve C over K with a K-rational point has infinitely many K-rational points. This conjecture would have some consequences in mathematical logic. For instance, it would imply the expected classification of stable fields of positive characteristic. We give some weak evidence for this conjecture. Specifically, suppose that K is a perfect field and Gal(K) is prime to p. Then every abelian variety A over K has Zariski dense K-rational points, and if C is a smooth curve of genus less than 7−p with at least one K-rational point, then C has infinitely many K-rational points.
Related
- Multiplicator freeness for restricted-ramification Galois groups over global function fields
- Abelian dynamical Galois groups over global function fields
- Finite subgroups of outer automorphism groups of the absolute Galois groups of mixed-characteristic local fields
- Explicit equations of Galois subfields of Hermitian function fields with respect to decomposition groups
- Galois groups of random polynomials of large degree
BibTeX
@misc{johnson2026conjecture,
title = {{A conjecture on Galois groups and large fields}},
author = {Will Johnson},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.41843},
eprint = {2610.41843},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.41843},
}
RIS
TY - UNPB
TI - A conjecture on Galois groups and large fields
AU - Will Johnson
PY - 2026
DA - 2026/10/30
PB - arXiv
DO - 10.48550/arxiv.2610.41843
UR - https://arxiv.org/abs/2610.41843
AB - Let p be a prime number and let K be a field such that p is prime to the absolute Galois group Gal(K), in the sense that p does not divide the degree of any finite Galois extension of K. We conjecture that K is large in the sense of Pop: any smooth algebraic curve C over K with a K-rational point has infinitely many K-rational points. This conjecture would have some consequences in mathematical logic. For instance, it would imply the expected classification of stable fields of positive characteristic. We give some weak evidence for this conjecture. Specifically, suppose that K is a perfect field and Gal(K) is prime to p. Then every abelian variety A over K has Zariski dense K-rational points, and if C is a smooth curve of genus less than 7−p with at least one K-rational point, then C has infinitely many K-rational points.
ER -