By Fred Diamond, Payman L. Kassaei, Minhyong Kim

ISBN-10: 1107693632

ISBN-13: 9781107693630

Automorphic kinds and Galois representations have performed a significant position within the improvement of contemporary quantity concept, 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 was once to discover fresh advancements during this quarter. The expository articles and study papers around the volumes mirror contemporary curiosity in p-adic tools in quantity concept and illustration idea, in addition to contemporary development on issues from anabelian geometry to p-adic Hodge concept and the Langlands application. the subjects coated in quantity comprise curves and vector bundles in p-adic Hodge thought, associators, Shimura kinds, the birational part conjecture, and different subject matters of latest curiosity.

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**Example text**

Set Div+ (D∗ ) = D = m x [x] | m x ∈ N, ∀I ⊂]0, 1[ x∈|D∗ | compact supp(D) ∩ D I is finite the monoid of effective divisors on D∗ . There is an exact sequence div 0 → B \ {0}/B× −−→ Div+ (D∗ ) −→ Pic+ (D∗ ) −→ 0. We are thus led to the question: for D ∈ Div+ (D∗ ), does there exist f ∈ B\{0} such that div( f ) = D ? Vector bundles on curves and p-adic Hodge theory 23 This is of course the case if supp(D) is finite. Suppose thus it is infinite. We will suppose moreover F is algebraically closed (the discrete valuation case is easier but this is not the case we are interested in) and thus |D∗ | = m F \ {0} where m F is the maximal ideal of O F .

14). Thus, u ∈ (Bb )ϕ=π d−d = 0 if d = d E if d = d . The surjectivity uses Weierstrass products. For this, let x ∈ WO E (O F ) be a primitive degree d element and D = div(x) its divisor. We are looking for f ∈ Pd \ {0} satisfying div( f ) = ϕ n (D). n∈Z Up to multiplying x by a unit we can suppose x ∈ π d + WO E (m F ). Then the infinite product + (x) = n≥0 ϕ n (x) πd Vector bundles on curves and p-adic Hodge theory 51 converges. For example, if x = π − [a], n + (π − [a]) = 1− n≥0 [a q ] . π One has div( + (x)) = ϕ n (D).

They know that for each integer n ≥ 1, OCm / p n OCm OC p / p n OC p but in a non canonical way. As a consequence of the preceding theorem we deduce almost étalness for characteristic 0 perfectoïd fields. 40. For L|E a perfectoïd field and L |L a finite extension we have m L ⊂ tr L |L (O L ). Proof. Set F = R(L ) and F = R(L). If L corresponds to m ∈ |Y F |, a = {x ∈ O F | |x| ≤ m } and a = {x ∈ O F | |x| ≤ m } we have identifications O L /πO L = O F /a O L /π O L = O F /a . According to point (2) of the preceding theorem, with respect to those identifications the map tr L |L modulo π is induced by the map tr F |F .

### Automorphic Forms and Galois Representations: Volume 2 by Fred Diamond, Payman L. Kassaei, Minhyong Kim

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