# Algebraic Curves and Projective Geometry by Edoardo Ballico, Ciro Ciliberto By Edoardo Ballico, Ciro Ciliberto

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Vector bundles on curves and p-adic Hodge theory 35 Point (4) is an easy consequence of the following classical characterization (0) n of ker θ: an element y = ∈ π A× is a n≥0 [yn ]π ∈ ker θ such that y0 generator of ker θ . In fact, if y is such an element then ker θ = (y) + π ker θ and one concludes ker θ = (y) by applying the π -adic Nakayama lemma (ker θ is π -adically closed). In point (3), the difficulty is to prove that the complete valued field L = A[ π1 ] is algebraically closed; other points following easily from point (2).

N∈Z If ρ = q −r ∈]0, 1] set |x|ρ = q −vr (x) . (3) For x = n [xn ]π n ∈ Bb set Newt(x) = decreasing convex hull of {(n, v(xn ))}n∈Z . In the preceding definition one can check that the function vr does not depend on the choice of a uniformizing element π . 1. For x ∈ Bb the function r → vr (x) defined on ]0, +∞[ is the Legendre transform of Newt(x). One has v0 (x) = lim vr (x). The Newton polygon of r→0 x is +∞ exactly on ] − ∞, vπ (x)[ and moreover lim Newt(x) = v0 (x). One +∞ has to be careful that since the valuation of F is not discrete, this limit is not always reached, that is to say Newt(x) may have an infinite number of strictly positive slopes going to zero.

2) Bb = {x ∈ B | Newt(x) is bounded below and ∃A, Newt(x)|]−∞,A] = +∞}. (3) The algebra {x ∈ B | ∃ A, Newt(x)|]−∞,A] = +∞} is a subalgebra of v(x n ) n WO E (F)[ π1 ] equal to n −∞ [x n ]π | liminf n ≥ 0 . n→+∞ This has powerful applications that would be difficult to obtain without Newton polygons. For example one obtains the following. 15. × (1) B× = Bb = x ∈ Bb | Newt(x) has 0 as its only non infinite slope . d (2) One has Bϕ=π = 0 for d < 0, Bϕ=Id = E and for d ≥ 0, Bϕ=π = B+ d ϕ=π d . Typically, the second point is obtained in the following way. 