By Fred Diamond, Payman L. Kassaei, Minhyong Kim

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

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**Example text**

H 1 G F , O F = 0. Using Tate’s method ([31]) this theorem is a consequence of the following “almost etalness” statement whose proof is much easier than in characteristic 0. 32. If L|F is a finite degree extension then m F ⊂ tr L/F (O L ). Sketch of proof. The trace tr L/F commutes with the Frobenius ϕ = Frobq . Choosing x ∈ O L such that tr L/F (x) = 0 one deduces that lim |tr L/F (ϕ −n (x))| = 1. 4) where 1 + WO E m F = ker WO E O F × −→ WO E k F × . 32) one has WO E O F GF = WO E (O F ). 33.

Suppose E = Q p and choose ρ ∈ |F × |∩]0, 1[. Let a ∈ F such that |a| = ρ. D. hull of the ideal W (O F )[a] of W (O F ) ⊗Z p Q p . This depends only on ρ since W (O F )[a] = {x ∈ W (O F ) | |x|0 ≥ ρ}. We thus have 1 [a n ] B+ . cris,ρ = W (O F ) n! n≥1 p One has + + B+ ρ p ⊂ Bcris,ρ ⊂ Bρ p−1 . From this one deduces ϕ n B+ cris,ρ , B+ = n≥0 + the ring usually denoted “Brig ” in p-adic Hodge theory. The ring B+ cris,ρ appears naturally in comparison theorems where it has a natural interpretation in terms of crystalline cohomology.

A+b=x We have the formula L (ϕ1 ∗ ϕ2 ) = L (ϕ1 ) + L (ϕ2 ). One can think of the Legendre transform as being a “tropicalization” of the Laplace transform: tropicalization (R, +, ×) −−−−−−−−→ (R, inf, +) Laplace transform /o /o /o / Legendre transform usual convolution ∗ /o /o o/ / tropical ∗ just defined. The function ϕ is a polygon, that is to say piecewise linear, if and only if L (ϕ) is a polygon. Moreover in this case: • the slopes of L (ϕ) are the x-coordinates of the breakpoints of ϕ, • the x-coordinates of the breakpoints of L (ϕ) are the slopes of ϕ.