The geometrically m-step solvable Grothendieck conjecture for genus 0 curves over finitely generated fields
Naganori Yamaguchi
TL;DR
This work advances anabelian understanding of genus $0$ hyperbolic curves over finitely generated fields by proving the geometrically $3$-step solvable Grothendieck conjecture in arbitrary characteristic. It develops a robust group-theoretic pipeline: (i) constructively reconstruct inertia and decomposition groups at cusps from higher-step solvable quotients using Blanchfield–Lyndon theory for free pro-$C$ groups and a weight filtration, (ii) define maximal cyclic subgroups of cyclotomic type to identify cusp data, and (iii) apply a rigidity invariant together with Frobenius-twist analysis to treat both characteristic $0$ and positive characteristic cases. The main results show that isomorphisms of $m$-step solvable quotients of the étale fundamental group determine the geometric isomorphism class of the curves up to twist-equivalence, with precise statements depending on the characteristic and parity of the data. The paper thus establishes a concrete, higher-step solvable proxy for Grothendieck’s anabelian program in genus $0$, with significant implications for understanding how solvable quotients encode cusp and moduli information. It introduces new tools—such as the rigidity invariant and the cyclotomic-type maximal cyclic subgroups—and leverages Galois descent to connect group-theoretic data with geometric structures. In sum, the results illuminate how solvable quotients retain enough arithmetic and geometric information to recover genus $0$ curves up to well-understood twists.
Abstract
In this paper, we present some partial results for the geometrically m-step solvable Grothendieck conjecture in anabelian geometry. Among other things, we prove the geometrically 3-step solvable Grothendieck conjecture for genus 0 curves over fields finitely generated over the prime field of arbitrary characteristic.