By Fred Diamond, Payman L. Kassaei, Minhyong Kim

ISBN-10: 1107691923

ISBN-13: 9781107691926

Automorphic kinds and Galois representations have performed a relevant position within the improvement of contemporary quantity idea, 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 varieties and Galois Representations' in July 2011, the purpose of which was once to discover contemporary advancements during this quarter. The expository articles and learn papers around the volumes replicate contemporary curiosity in p-adic tools in quantity conception and illustration idea, in addition to contemporary growth on subject matters from anabelian geometry to p-adic Hodge concept and the Langlands application. the themes lined in quantity one contain the Shafarevich Conjecture, powerful neighborhood Langlands correspondence, p-adic L-functions, the basic lemma, and different subject matters of latest curiosity.

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If W is a closed subspace of V ∗ stable under G, then let W ⊥ = {v ∈ V such that f (v) = 0 for all f ∈ W }. 16. Moreover, W ⊥ is a closed subspace of V , that is also stable under G, so that either W ⊥ = {0} and W = V ∗ or W ⊥ = V and W = {0}. Assume now that G is a topologically finitely generated profinite group (in this chapter, we only need the case G = Z p ). Denote by V (G) the subE-vector space of V generated by the elements (g − 1)v where g ∈ G and v ∈ V . 18. If V ∈ Veccomp (E), then V (G) is a closed subspace of V .

As a first step, we prove that Breuil’s category of filtered S-modules S p−1 can be replaced by a similar category L f of free filtered W-modules (M, F (M)) with σ -linear maps ϕ : F(M) −→ M and differentiations N : M −→ M ⊗W S. Then we define a torsion analogue Lt of the category L f . 1. Note that Lt contains the full subcategory L f t whose objects are subquotients of objects of L f and this subcategory is strictly smaller than Lt . This is very special feature of “semi-stable” theory: t if we start with the subcategory S cr p−1 then the appropriate categories Lcr and ft Lcr coincide.

0 Note that the correspondence p p p [r0 mod x 0 ]T1 + [r0 ] + p[r1 ] → (r0 + x0 r1 ) mod x0 m R ∗ determines an epimorphic map A0cr,2 / pA0cr,2 −→ R0 in the category L0 and this map induces isomorphism of K -modules V f t (L[γ ]) and V ∗ (L[γ ]). 3. Properties of modified functor The following property was our main target. 7. CV ft is fully faithful. Proof. By devissage it will be sufficient to verify this statement on the level of the subcategories of killed by p objects. The corresponding restricft tion of CV is equivalent then to the functor CV ∗ from Section 2 (cf.

### Automorphic Forms and Galois Representations: Volume 1 by Fred Diamond, Payman L. Kassaei, Minhyong Kim

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