By Fred Diamond, Payman L. Kassaei, Minhyong Kim

Automorphic types and Galois representations have performed a valuable position within the improvement of contemporary quantity conception, with the previous coming to prominence through the prestigious Langlands software and Wiles' evidence of Fermat's final Theorem. This two-volume assortment arose from the 94th LMS-EPSRC Durham Symposium on 'Automorphic kinds and Galois Representations' in July 2011, the purpose of which was once to discover contemporary advancements during this region. The expository articles and examine papers around the volumes mirror contemporary curiosity in p-adic equipment in quantity conception and illustration idea, in addition to fresh growth on themes from anabelian geometry to p-adic Hodge idea and the Langlands application. the subjects lined in quantity contain curves and vector bundles in p-adic Hodge concept, associators, Shimura kinds, the birational part conjecture, and different issues of latest curiosity.

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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.