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

Naganori Yamaguchi

TL;DR

This work develops and extends the geometrically $m$-step solvable Grothendieck conjecture for affine hyperbolic curves over fields finitely generated over the prime field. It proves weak and strong bi-anabelian forms over finite fields and finitely generated fields by replacing full profinite fundamental groups with their $m$-step solvable quotients, and by reconstructing curve data from these quotients via weight filtrations, inertia/decomposition data, and class-field-theoretic arguments. A key achievement is the $m$-step version of the Oda–Tamagawa good reduction criterion, including descent refinements to non-perfect residue fields, and the development of Frobenius-localized categories to handle finitely generated bases. The results show that the $m$-step tame fundamental group $ ext{Pi}_U^{(m)}$ determines the hyperbolic curve $U$ up to isomorphism in the appropriate base category, enabling concrete reconstruction and lifting results for both finite-field and finitely generated-field settings. Overall, the paper advances anabelian geometry by anchoring curve reconstruction in lower-depth, computable quotients of the tame fundamental group with precise conditions for lifting isomorphisms to geometric identifications, broadening the scope and applicability of Grothendieck’s conjectures in arithmetic geometry.

Abstract

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.

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

TL;DR

This work develops and extends the geometrically -step solvable Grothendieck conjecture for affine hyperbolic curves over fields finitely generated over the prime field. It proves weak and strong bi-anabelian forms over finite fields and finitely generated fields by replacing full profinite fundamental groups with their -step solvable quotients, and by reconstructing curve data from these quotients via weight filtrations, inertia/decomposition data, and class-field-theoretic arguments. A key achievement is the -step version of the Oda–Tamagawa good reduction criterion, including descent refinements to non-perfect residue fields, and the development of Frobenius-localized categories to handle finitely generated bases. The results show that the -step tame fundamental group determines the hyperbolic curve up to isomorphism in the appropriate base category, enabling concrete reconstruction and lifting results for both finite-field and finitely generated-field settings. Overall, the paper advances anabelian geometry by anchoring curve reconstruction in lower-depth, computable quotients of the tame fundamental group with precise conditions for lifting isomorphisms to geometric identifications, broadening the scope and applicability of Grothendieck’s conjectures in arithmetic geometry.

Abstract

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.
Paper Structure (14 sections, 57 theorems, 56 equations)

This paper contains 14 sections, 57 theorems, 56 equations.

Table of Contents

  1. Basic results on $\Pi_{U}^{(m)}$
  2. The center-freeness of $\Pi_{U}^{(m,\Sigma)}$.
  3. The group-theoretical reconstruction of various invariants of $\Pi_{U}^{(m,\Sigma)}$.
  4. Inertia groups of $\overline{\Pi}^{m,\Sigma}_{U}$
  5. The case of finite fields
  6. The separatedness of decomposition groups of $\Pi_{U}^{(m)}$
  7. The group-theoretical reconstruction of decomposition groups of $\Pi_{U}^{(m)}$
  8. The weak bi-anabelian results over finite fields
  9. The strong bi-anabelian results over finite fields
  10. The $m$-step solvable version of the good reduction criterion for hyperbolic curves
  11. The case of finitely generated fields
  12. The category $\text{Sch}_{k}^{\text{geo.red.}}$
  13. The weak bi-anabelian results over finitely generated fields
  14. The strong bi-anabelian results over finitely generated fields

Key Result

Theorem 3

Assume that $m\geq 2$ and that $k$ is an algebraic number field satisfying one of the following conditions (a)-(b). Let $\lambda_{i}$ be an element of $k-\{0,1\}$ and set $\Lambda_{i}:=\{0,1,\infty,\lambda_{i}\}$ for each $i=1,2$. Then the following holds.

Theorems & Definitions (122)

  • Conjecture 1: The (relative, geometrically) $m$-step solvable Grothendieck conjecture
  • Remark 2
  • Theorem 3: cf. Na1990-405 Theorem A
  • Theorem 4: cf. Mo1999 Theorem A${'}$
  • Remark 5
  • Theorem 6: cf. Ya2020 Theorem 2.4.1
  • Theorem 7: Theorem \ref{['finGCweak']}, Corollary \ref{['finGCcorinn']}
  • Theorem 8: Theorem \ref{['fingeneGCweak']}, Corollary \ref{['fingeneGCcorinn2']}
  • Theorem : Summary of new results contained in Theorem \ref{['ccc']}
  • Theorem 9: Theorem \ref{['2goodreductiontheorem']}
  • ...and 112 more