On a p-adic version of Narasimhan and Seshadri's theorem

2024/06/18 by Fabrizio Andreatta, Andreatta, Fabrizio
Mathematics · #Advanced Mathematical Identities #Algebraic Geometry (math.AG) #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT) #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.2406.12766

openalex publication_date 2024/06/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Consider a smooth projective curve C of genus g over a complete discrete valuation field of characteristic 0 and residue field \Fbar_p. Motivated by Narasimhan and Seshadri's theorem, Faltings asked whether all semistable vector bundles of degree 0 over C_\Cp are in the image of the p-adic Simpson correspondence. Works of Deninger-Werner and Xu show that this is equivalent for the vector bundle to having potentially strongly semistable reduction. We prove that if C has good reduction, p>r(r-1) (g-1) and we consider a vector bundle of rank r with stable reduction, the conditions of having potentially strongly semistable reduction and of having strongly semistable reduction are equivalent. In particular, we provide a negative answer to Faltings' question

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