Minima successifs des réseaux et pentes des fibrés vectoriels sur les corps de fonctions globaux

TL;DR

The paper develops a precise dictionary between normed $A$-lattices on a global function field and vector bundles on the base curve, and uses this to relate the Minkowski successive minima $\lambda_i(\overline{M})$ to Harder–Narasimhan slopes $\mu_i(E^{\overline{M}})$. It provides effective structural results: a decomposition of an $A$-lattice into rank-1 submodules with controlled index (Mahler-type reduction) and explicit bounds tying minima to slopes via constants depending only on the genus $\mathfrak g$, the place degree $\mathfrak f$, and the rank $n$. In the genus-zero, degree-one case these relations become equalities, recovering the classical correspondences for function fields, while in general they give quantitative bridges between diophantine-style lattice data and geometric slope theory. The results illuminate the function-field analogue of reduction theory and feed into applications in arithmetic geometry and parametric number theory on function 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.