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Naganori Yamaguchi

Publications and source records attributed to Naganori Yamaguchi.

9 recordsLinked to original sources
arXiv2026

On branched coverings of the projective line over the integers

We investigate the étale fundamental group of the complement of a horizontal divisor on $\mathbb{P}^{1}_{\mathbb{Z}}$. We prove that this group has no nontrivial finite solvable quotient if and only if the divisor is normal crossings at the prime~$2$. Moreover, if the divisor is normal crossings at the prime~$2$ and either has three irreducible components or is normal crossings at the prime~$3$, we show that no quotient isomorphic to $\mathrm{PSL}_{2}(q)$ can occur for certain prime powers~$q$.

math.AG↗
arXiv2026

The metabelian Grothendieck conjecture for genus zero curves over finitely generated fields

In this paper, we prove the metabelian Grothendieck conjecture for genus-zero curves over finitely generated fields. More precisely, we show that two hyperbolic genus-zero curves are isomorphic over the base field, up to Frobenius twist in positive characteristic, if and only if their geometrically maximal metabelian tame fundamental groups are isomorphic over the absolute Galois group of the base field.

math.AG↗
arXiv2026

Étale fundamental groups of smooth arithmetic surfaces and the Grothendieck conjecture

We study the structure of the étale fundamental groups of smooth curves over certain arithmetic schemes, and investigate the relative version of Grothendieck's anabelian conjecture in this setting. Consequently, every hyperbolic curve over the ring of S-integers of a number field in which a rational prime is inverted is anabelian, i.e., its schematic structure is completely determined by its étale fundamental group. Moreover, we obtain a partial result toward the semi-absolute version of Grothendieck's anabelian conjecture in this context.

math.NT↗
arXiv2026

Center-freeness of finite-step solvable groups arising from anabelian geometry

Anabelian geometry suggests that, for suitably geometric objects, their étale fundamental groups determine the geometric objects up to isomorphism. From a group-theoretic viewpoint, this philosophy requires rigidity properties, which often follow from their center-freeness of the associated étale fundamental groups. In fact, some profinite groups arising from anabelian geometry are center-free. For any integer $m\geq 2$, we investigate how such center-freeness behaves under passage to the maximal $m$-step solvable quotients. In particular, we show that the maximal $m$-step solvable quotients of the étale and tame fundamental groups of a hyperbolic curve over a separably closed field are torsion-free and center-free. Furthermore, we show that this implies the rigidity property of the $m$-step solvable Grothendieck conjecture.

math.GR↗
arXiv2026

Anabelian geometry for Deligne-Mumford curves

We develop an anabelian framework for general Deligne-Mumford curves, showing that their stack and orbifold structures are encoded in the group-theoretic properties of their étale fundamental groups. After establishing the required properties for profinite F-groups, we prove that fundamental geometric features, including hyperbolicity, affineness, and inertia data, can already be detected from low-level solvable quotients of the associated profinite groups, namely at the optimal 3-step level. As a consequence, we obtain some anabelian reconstruction results for Deligne-Mumford curves, their rigidifications, and their coarsification. While the m-step Grothendieck conjecture doesn't hold for Deligne-Mumford curves, we establish a 5-step anabelian theorem for the rigidification of affine Deligne-Mumford curves, namely affine stacky curves. A certain emphasis is given to the role of stack inertia groups.

math.AG↗
arXiv2025

Symmetries of spaces and numbers -- anabelian geometry

``Can number and geometric spaces be reconstructed from their symmetries?'' This question, which is at the heart of anabelian geometry, a theory built on the collaborative efforts of an international community in many variants and with the Japanese arithmetic school as a core, illustrates, in the case of a positive answer, the universality of the homotopic method in arithmetic geometry. Starting with elementary examples, we first introduce the motivations and guiding principles of the theory, then present its most structuring results and its contemporary trends. As a result, the reader is presented with a rich and diverse landscape of mathematics, which thrives on theoretical and explicit methods, and runs from number theory to topology.

math.NT↗
arXiv2023

The geometrically m-step solvable Grothendieck conjecture for affine hyperbolic curves over finitely generated fields

In this paper, we present some new results on the geometrically m-step solvable Grothendieck conjecture in anabelian geometry. Specifically, we show the (weak bi-anabelian and strong bi-anabelian) geometrically m-step solvable Grothendieck conjecture(s) for affine hyperbolic curves over fields finitely generated over the prime field. First of all, we show the conjecture over finite fields. Next, we show the geometrically m-step solvable version of the Oda-Tamagawa good reduction criterion for hyperbolic curves. Finally, by using these two results, we show the conjecture over fields finitely generated over the prime field.

math.AG↗