Research papers
This collection develops selected results from three papers on Galois groups and coverings over the integers. The first asks whether finite Galois automorphisms can be distinguished through their actions on abelian quotients, and proves this form of faithfulness for discretely valued fields of mixed characteristic and for higher local fields of positive dimension.
The second explains when removing a horizontal divisor leaves solvable covering theory unchanged. Under its arithmetic boundary condition, it also reduces the extension of a representation across the divisor to the same question for its reduction modulo the maximal ideal.
The third studies complements of rational sections of the projective line. Determinant valuations measure intersections and detect configurations that force a connected double cover, including every configuration with at least seven sections. These results establish that quadratic obstruction; the paper's full equivalence between normal crossings at two and prosolvable triviality is not yet part of the selection.
Selected theorems and definitions
- definition
Ab-faithfulness
Finite relative quotients must be detectable by their action on abelianized open normal subgroups.
- theorem
Mixed-characteristic discrete valuations
Conjugation on the mod-p abelianized absolute Galois group detects every element of a finite Galois group.
- theorem
Higher local fields
Every positive-dimensional higher local field has an ab-faithful absolute Galois group.
- theorem
The étale fundamental group of Spec Z
Every geometric base point gives the trivial étale fundamental group of the integers.
- definition
The arithmetic boundary condition
A local condition at characteristic two is combined with a geometric-integrality condition that survives every connected finite étale cover.
- theorem
A perfect boundary kernel
Under the arithmetic boundary condition, removing the divisor does not change the maximal prosolvable fundamental group.
- theorem
Extending a representation from its reduction
A continuous representation extends across the boundary exactly when its residue representation does.
- theorem
Normal crossings from rational sections
Normal crossings can be tested by counting colliding sections and measuring the valuation of their determinants.
- theorem
Intersection multiplicity as a determinant valuation
The local intersection length of two rational sections is exactly the p-adic valuation of their determinant.
- theorem
A quadratic quotient from failure at 2
Failure of normal crossings over the 2-adic integers produces a nontrivial quotient of the actual arithmetic fundamental group.
- theorem
At least seven sections force a quotient
The complement of seven or more distinct rational sections always has a cyclic quotient of order two.
References used in these guides
- Naganori Yamaguchi. Ab-faithfulness of absolute Galois groups in number theory. 2026. arXiv:2610.05634v1, submitted October 4, 2026.
- Naganori Yamaguchi. Extension of solvable coverings across horizontal divisors over the integers. 2026. arXiv:2610.02948v1, submitted October 2, 2026.
- Ryoji Shimizu and Naganori Yamaguchi. On branched coverings of the projective line over the integers. 2026. arXiv:2608.15057v1, submitted August 15, 2026.
- Alexander Grothendieck et al.. Revêtements étales et groupe fondamental (SGA 1). Springer, Lecture Notes in Mathematics 224, 1971. Link to the updated edition with remarks by Michel Raynaud.