Semi-stable extensions on arithmetic surfaces

Abstract

Given a line bundle L on a smooth projective curve over the complex numbers, we show that a general extension E of L by the trivial line bundle is very stable: line bundles contained in E have degree much less than half the degree of E. From this result we deduce new inequalities for the successive minima of the euclidean lattice H1(X,L−1), where L is an hermitian line bundle on the arithmetic surface X.