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Logarithmic geometry and Frobenius, II

Kazuya Kato, Chikara Nakayama, Sampei Usui

TL;DR

This work develops an l-adic analogue of SL(2)-orbit theorems via the notion of monodromy systems, establishing a parallel between log mixed Hodge theory and ℓ-adic weight filtrations. It introduces ratio spaces and torus twists to control asymptotics of monodromy data, proving refined orbit theorems for multi-variable and one-variable cases and their extensions to the δ_W and spl_W invariants. The results yield asymptotic formulas for regulators and local height pairings on non-archimedean fields and provide a concrete verification of the Part I expectation in a degenerate Tate elliptic curve example. Collectively, the paper bridges Hodge-theoretic and ℓ-adic/logarithmic frameworks, offering new tools for understanding degenerations and arithmetic invariants through SL(2)-orbit techniques.

Abstract

Based on the strong analogy between the category of log mixed Hodge structures and the category ${\cal A}_X$ of $\ell$-adic nature, which we have introduced in the previous part and is closely related to the weight-monodromy conjecture, we prove the $\ell$-adic analogues of some theorems in Hodge theory related to the SL(2)-orbit theorem.

Logarithmic geometry and Frobenius, II

TL;DR

This work develops an l-adic analogue of SL(2)-orbit theorems via the notion of monodromy systems, establishing a parallel between log mixed Hodge theory and ℓ-adic weight filtrations. It introduces ratio spaces and torus twists to control asymptotics of monodromy data, proving refined orbit theorems for multi-variable and one-variable cases and their extensions to the δ_W and spl_W invariants. The results yield asymptotic formulas for regulators and local height pairings on non-archimedean fields and provide a concrete verification of the Part I expectation in a degenerate Tate elliptic curve example. Collectively, the paper bridges Hodge-theoretic and ℓ-adic/logarithmic frameworks, offering new tools for understanding degenerations and arithmetic invariants through SL(2)-orbit techniques.

Abstract

Based on the strong analogy between the category of log mixed Hodge structures and the category of -adic nature, which we have introduced in the previous part and is closely related to the weight-monodromy conjecture, we prove the -adic analogues of some theorems in Hodge theory related to the SL(2)-orbit theorem.

Paper Structure

This paper contains 9 sections, 41 theorems, 71 equations.

Table of Contents

  1. Monodromy systems
  2. Ratio spaces
  3. Analogy
  4. SL(2)-orbit theorems on the torus twist back
  5. SL(2)-orbit theorems in one variable
  6. SL(2)-orbit theorems on ${\rm{spl}}_W$
  7. SL(2)-orbit theorems on $\delta_W$
  8. Asymptotic behaviors of regulators and local height pairings
  9. On the expectation 6.16 of Part I

Key Result

Theorem 2

(Cf. KNU08 Theorem $0.5$.) Let $(N_1, \dots, N_n,F)$ be a nilpotent orbit. Then for $y_j>0$ ($1\leq j\leq n$), $y_j/y_{j+1}\gg 0$ ($y_{n+1}$ denotes $1$), ${\rm{spl}}_W(\exp(\sum_{j=1}^n iy_jN_j)F)$ is expressed by convergent Taylor series in $y_{j+1}/y_j$ ($1\leq j\leq n$).

Theorems & Definitions (44)

  • Theorem 2
  • Theorem 3
  • Definition 1.2
  • Lemma 1.5
  • Proposition 1.6
  • Corollary 1.7
  • Proposition 1.9
  • Lemma 1.10
  • Lemma 1.11
  • Proposition 1.17
  • ...and 34 more