Minima successifs des r\'eseaux et pentes des fibr\'es vectoriels sur les corps de fonctions globaux

TL;DR

This paper explores the relationship between Mahler successive minima of normed $A$-lattices and the Harder-Narasimhan slopes of vector bundles over a smooth projective curve over a finite field, revealing deep connections in algebraic geometry.

Contribution

It establishes precise links between Mahler minima and Harder-Narasimhan slopes via category equivalence, advancing understanding of vector bundles over algebraic curves.

Findings

Established relationships between Mahler minima and vector bundle slopes

Connected lattice theory with algebraic geometry concepts

Provided a framework for analyzing vector bundles over finite fields

Abstract

Let ${\bf C}$ be a smooth geometrically connected projective curve over a finite field, and let $A$ be the affine algebra of its regular functions outside a fixed place of ${\bf C}$. We give precise relationships between the Mahler successive minima of normed $A$-lattices and the Harder-Narasimhan slopes of vector bundles over ${\bf C}$ using their category equivalence.