The strong bi-anabelian m-step solvable Grothendieck conjecture

Let $k$ be a field with absolute Galois group $G_k$ , let $U_1,U_2$ be the curves under consideration, and let $m,n$ be non-negative integers. For each $r$ , write $\pi^{\acute{e}\mathrm{t}}_1(U_i,\ast)^{(r)}$ for the corresponding geometrically maximal $r$ -step solvable quotient. The groups of $k$ -isomorphisms and $G_k$ -equivariant isomorphisms are denoted by $\mathrm{Isom}_k$ and $\mathrm{Isom}_{G_k}$ , respectively, and $\mathrm{Inn}$ denotes inner automorphisms. The strong bi-anabelian form of the $m$ -step solvable Grothendieck conjecture. Fix a non-negative integer $n$ . The image of

\mathrm{Isom}_{k}(U_{1},U_{2})\to \mathrm{Isom}_{G_{k}}(\pi^{\acute{e}\mathrm{t}}_{1}(U_{1},\ast)^{(m)},\pi^{\acute{e}\mathrm{t}}_{1}(U_{2},\ast)^{(m)})/\mathrm{Inn}(\pi^{\acute{e}\mathrm{t}}_{1}(U_{2,k^{\mathrm{sep}}},\ast)^{m})

coincides with the image of

\mathrm{Isom}_{G_{k}}(\pi^{\acute{e}\mathrm{t}}_{1}(U_{1},\ast)^{(m+n)},\pi^{\acute{e}\mathrm{t}}_{1}(U_{2},\ast)^{(m+n)})/\mathrm{Inn}(\pi^{\acute{e}\mathrm{t}}_{1}(U_{2,k^{\mathrm{sep}}},\ast)^{m+n}).

The strong form refines the weak form by incorporating functoriality. The source discusses previous results and the role of coverings, but does not resolve the conjecture in this general formulation.

The conjecture asks whether isomorphisms between sufficiently deep solvable quotients of the étale fundamental groups determine exactly the isomorphisms of the curves, with functorial compatibility. It is presented in the 2022 survey as the strong bi-anabelian refinement of the $m$ -step solvable Grothendieck conjecture.

Known results

A 2023 paper reports both weak and strong bi-anabelian results for affine hyperbolic curves over fields finitely generated over the prime field, using finite-field results and a solvable Oda–Tamagawa criterion.
Under substantial hypotheses including $m\ge 5$ , $n\ge 3$ , and $m>n$ , it reports bijectivity of the relevant curve-to-fundamental-group map.
A 2022 survey records proof sketches for related theorems of H. Nakamura, S. Mochizuki, and its author, but does not claim the unrestricted conjecture is solved.

2024 genus- $0$ development

A 2024 paper proves a genus- $0$ result for the weak bi-anabelian form over finitely generated fields and explicitly says that its strong-form consequences are contained in earlier work; it does not present a new general proof of the stated conjecture.

Current status (as of August 2026): The unrestricted strong conjecture remains open; strong-form results are reported only under substantial hypotheses, and these reports are not independently verified here.

The strong bi-anabelian m-step solvable Grothendieck conjecture

Primary source

Progress summary

Solutions 0