Very stable extensions on arithmetic surfaces

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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.