Tropical reduction and lifting of q-differentials on Berkovich curves

2021/09/22 by Uri Brezner, Brezner, Uri
Computer Science · Mathematics · #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Polynomial and algebraic computation

paper · pdf · doi:10.48550/arxiv.2109.10628

openalex publication_date 2021/09/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Given a complete real-valued field k of residue characteristic zero, we study properties of a meromorphic q-differential form ξ (a section of Ω_X^{⊗ q}) on a smooth proper k-analytic curve X. In particular, we associate to (X,ξ) a natural tropical reduction datum combining tropical-geometric data over the value group of |k^×| and algebro-geometric reduction data over the residue field \widetildek. We show that this datum satisfies natural compatibility conditions, and prove a lifting theorem asserting that any compatible tropical reduction datum lifts to an actual pair (X, ξ). This generalizes the result of \citeTT20 from q=1 to a general natural number q. Furthermore, it is a non-Archimedean analog of \citeBCGGM16.

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