By Fred Diamond, Payman L. Kassaei, Minhyong Kim

Automorphic kinds and Galois representations have performed a relevant position within the improvement of recent quantity concept, with the previous coming to prominence through the distinguished 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 zone. The expository articles and learn papers around the volumes mirror contemporary curiosity in p-adic tools in quantity idea and illustration idea, in addition to fresh development on themes from anabelian geometry to p-adic Hodge conception and the Langlands software. the themes lined in quantity comprise curves and vector bundles in p-adic Hodge concept, associators, Shimura kinds, the birational part conjecture, and different subject matters of latest curiosity.

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**Extra resources for Automorphic Forms and Galois Representations: Volume 2**

**Sample text**

27. For ∈ m F \ {0} define u = [ [ ]Q ∈ WO E (O F ). 1/q ] Q This is a primitive degree one element since it is equal to the power series evaluated at [ 1/q ] Q . For example, if E = Q p and LT = Gm , setting = 1 + one has Q(T ) T 1 p u =1+ + ··· + p−1 p . 28. There is a bijection ∼ G(O F ) \ {0} /O× E −→ |Y | O× E . −→ (u ). The inverse of this bijection is given by the following rule. xn+1 = xn . More generally, X (G) will stand for the projective limit “lim G” where the ←− n≥0 transition mappings are multiplication by π (one can give a precise geometric meaning to this but this is not our task here, see [12] for more details).

Define Div+ (Y/ϕ Z ) = {D ∈ Div+ (Y ) | ϕ ∗ D = D}. There is an injection |Y |/ϕ Z −→ Div+ (Y /ϕ Z ) m −→ [ϕ n (m)] n∈Z that makes Div+ (Y/ϕ Z ) a free abelian monoid on |Y |/ϕ Z . If x ∈ Bϕ=π \ {0} then div(x) ∈ Div+ (Y /ϕ Z ). 58. If F is algebraically closed the morphism of monoids Pd \ {0} /E × −→ Div+ (Y/ϕ Z ) div : d≥0 is an isomorphism. Let us note the following important corollary. 59. If F is algebraically closed the graded algebra P is graded factorial with irreducible elements of degree 1.

Thus, Y /ϕ Z should have a sense as a “rigid” space. Let’s look in more details at what this space Y /ϕ Z should be. Vector bundles on curves and p-adic Hodge theory 49 It is easy to classify rank 1 ϕ-modules over B. They are parametrized by Z: to n ∈ Z one associates the ϕ-module with basis e such that ϕ(e) = π n e. We thus should have ∼ Z −→ Pic(Y/ϕ Z ) n −→ L ⊗n where L is a line bundle such that for all d ∈ Z, H 0 (Y/ϕ Z , L ⊗d ) = Bϕ=π . d If E = Fq ((π )) and F is algebraically closed Hartl and Pink classified in [20] the ϕ-modules over B, that is to say ϕ-equivariant vector bundles on D∗ .