How far automorphisms of a p-adic absolute Galois group move the ring of integers: to the edge of the log-shell, with an application to Step (v) of IUT IV
Year: 2026
Type: preprint
Abstract: Let p be an odd prime and K/Qp a finite extension with ramification index e ≤ p − 2. A continuous automorphism α of the absolute Galois group GK induces, through the mono-anabelian reconstruction, a Qp-linear automorphism α+ of K. The map α+ preserves Λ = log U1K = 𝔪K and the log-shell 𝔪K1−e, but in general not the ring of integers 𝒪K (Hoshi). This note shows that for every integer m the Zp-module spanned by all images of 𝔪Km is 𝔪Klo(m), where lo(m) is the largest integer ≤ m congruent to 1 modulo e. So the largest drop of the valuation is (m − 1) mod e: the trivial upper bound coming from the invariance of Λ is attained. In particular the images of 𝒪K span the whole log-shell, and if e ≥ 2 some integral element is carried to its boundary. When [K:Qp] is even, or odd with residue degree at least 2, this follows from Kondo's description of the kernel of the trace and Kondo's transvections, from the preservation of traces shown by Hoshi–Nishio, and from automorphisms of free groups that mix the pairs of the Jannsen–Wingberg presentation. In the remaining case, odd degree with total ramification, everything is decided by whether one line in Λ/pΛ, the image of pZp, can be moved. To move it, an automorphism that shifts the generator x1 of the Jannsen–Wingberg presentation in the direction of x2 is constructed on the quotient of GK by the third term of the descending p-central series of the wild inertia group, and lifted to GK by Wingberg's lifting theorem (NSW (7.5.15)). As a by-product, for every finite extension k/Qp of odd degree d ≥ 3 there is an automorphism with α+(y1) ≠ y1, where y1 corresponds to the first generator. This answers the point left open in Remark 1.10 of Kondo, arXiv:2512.09231, whether such an element exists in odd degree. It also gives automorphisms that move (Qp)+ ⊂ k. As an application to the holomorphic hull in Step (v) of the proof of IUT IV, Theorem 1.10, the note shows that at small places the actual indeterminacies (Ind1), (Ind2) always reach the exact value found in the author's earlier note Three tiers of indeterminacy (doi:10.5281/zenodo.24043000); the hypotheses λ ∈ Z and (U) of that note are not needed. Among the bad places of the explicit initial Θ-datum D109, the one above 534775081 is of the odd-degree, totally ramified kind. Remaining questions are listed, among them the determination of the group of induced maps itself. The note credits the earlier work it builds on: the mono-anabelian reconstruction and the non-group-theoretic nature of the ring of integers (Hoshi), the preservation of traces (Hoshi–Nishio), the kernel of the trace, the transvections and the question of Remark 1.10 (Kondo), the presentation of Jannsen–Wingberg and the lifting theorem of Wingberg in the form of Neukirch–Schmidt–Wingberg, and the uses of the same lifting theorem by NSW and Nishio. On the IUT side it cites statements of Mochizuki, Joshi, Scholze–Stix, Dupuy–Hilado, Minamide–Tsujimura, lana-agents/iut, Takkun-kohinata/IUT_LEAN and Zeng. No claim is made about the truth of Corollary 3.12 of IUT III or of the abc conjecture. The proofs are on paper; nothing is formally verified. Python scripts check the identities in free groups, a determinant, and the group-ring computations of the construction in finite models. The question studied here arose in the author's discussions with generative AI tools, and the author decided to make it the subject of this note. The proofs were found with generative AI tools (large language models) under the author's direction, and these tools were also used for the computations, the literature search and the writing. The author's own reading of the proofs is still in progress. The author takes full responsibility for the note. The deposit contains the note and an archive of the folder log-shell-edge/ (LaTeX source, Python scripts and their outputs), made from tag v0.26.0 of github.com/turahigashi/iut-explicit-initial-theta.
Author: Urahigashi, Toshihisa
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225590661