By Fred Diamond, Payman L. Kassaei, Minhyong Kim

ISBN-10: 1107691923

ISBN-13: 9781107691926

Automorphic types and Galois representations have performed a principal position within the improvement of recent quantity thought, 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 types and Galois Representations' in July 2011, the purpose of which was once to discover fresh advancements during this zone. The expository articles and examine papers around the volumes mirror contemporary curiosity in p-adic tools in quantity conception and illustration thought, in addition to contemporary growth on subject matters from anabelian geometry to p-adic Hodge conception and the Langlands software. the subjects coated in quantity one comprise the Shafarevich Conjecture, powerful neighborhood Langlands correspondence, p-adic L-functions, the elemental lemma, and different themes of up to date curiosity.

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**Sample text**

The maximal abelian extension of Q in Qcr (5) equals Q(ζ25 ), where ζ25 is a 25-th primitive root of unity. Let L 2 be the maximal abelian extension of L 1 inside Qcr (5). The class number h(L 1 ) = 1 implies that L 2 is totally ramified at 5 over Q and L 2 /L 1 is a 5-extension. Because the total degree is √ 5 less than 340 we have [L 2 : L 1 ] 5 and one can see that L 2 = L 1 ( 2 + 5). Then the maximal upper ramification number of this field extension is 1 and the maximal lower number is 8, therefore, the normalized discriminant of L 2 ∗ , cf.

11. If Qst ( p) is totally ramified at p then there are p-divisible groups H = H0 ⊃ H1 ⊃ H2 in M Q such that H0 /H1 is the product of several copies of Q p /Z p , H2 is the product of several copies of (Q p /Z p )( p − 1) and all simple subquotients in H1 /H2 come from objects F j = (F p ( j ), 0, 0) with 1 j p − 2. 3. General criterion Suppose X/Q is a projective variety, p is a prime number and N ∈ N. Then V = HetN (X Q¯ , Q p ) is a finite dimensional Q p -vector space with continuous Q -action.

1. Breuil’s functor We work with Breuil’s theory of semi-stable p-adic representations of K = Gal( K¯ /K ), [14]–[16]. This theory allows us to construct K -invariant lattices in semi-stable Q p [ K ]-modules with Hodge–Tate weights from [0, p). 2. 1. 1. Note that Aˆ st is provided with the appropriate S-module Semi-stable case 19 structure, filtration, morphisms ϕ and N , and K -module structure. The notation Hom F,ϕ,N means the set of all S-linear homomorphisms compatible with filtrations and the morphisms ϕ and N .

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

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