By Fred Diamond, Payman L. Kassaei, Minhyong Kim

Automorphic types and Galois representations have performed a crucial position within the improvement of recent quantity thought, with the previous coming to prominence through the distinguished Langlands application and Wiles' facts 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 sector. The expository articles and study papers around the volumes mirror contemporary curiosity in p-adic equipment in quantity concept and illustration idea, in addition to fresh growth on issues from anabelian geometry to p-adic Hodge idea and the Langlands software. the themes lined in quantity contain curves and vector bundles in p-adic Hodge thought, associators, Shimura types, the birational part conjecture, and different subject matters of latest curiosity.

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R > 0 are some slopes of Newt(f) one can write r f = g. 1− i=1 [ai ] π where v(ai ) = λi . Using this one proves the following. 48. Suppose F is algebraically closed. For f ∈ B there exists a sequence (ai )i≥0 of elements of m F going to zero and g ∈ B[0,1[ such that +∞ f = g. 1− i=0 [ai ] . π If moreover f ∈ B+ there exists such a factorization with g ∈ WO E (O F ). 3. 49. Define Div+ (Y ) = am [m] ∀I ⊂]0, 1[ m∈|Y | compact {m | am = 0 and m ∈ I } is finite . Vector bundles on curves and p-adic Hodge theory 47 For f ∈ B \ {0} set ordm ( f )[m] ∈ Div+ (Y ).

For example, as a consequence, if a1 , . . , an ∈ m F \ {0}, then Newt (π − [a1 ]) . . (π − [an ]) is +∞ on ] − ∞, 0[, 0 on [n, +∞[ and has non-zero slopes v(a1 ), . . , v(an ). 5. Define • B = completion of Bb with respect to (| · |ρ )ρ∈]0,1[ , • B+ = completion of Bb,+ with respect to (| · |ρ )ρ∈]0,1[ , • for I ⊂]0, 1[ a compact interval B I = completion of Bb with respect to (| · |ρ )ρ∈I . The rings B and B+ are E-Frechet algebras and B+ is the closure of Bb,+ in B. Moreover, if I = [ρ1 , ρ2 ] ⊂]0, 1[, for all f ∈ B sup | f |ρ = sup{| f |ρ1 , | f |ρ2 } ρ∈I because the function r → vr ( f ) is concave.

Then: (1) This defines an ultrametric distance on |Y |deg=1 . (2) For any ρ ∈]0, 1[, |Y |deg=1, · ≥ρ , d is a complete metric space. 25. In equal characteristic, if E = Fq ((π )), then |Y | = |D∗ | = m F \{0} and this distance is the usual one induced by the absolute value |·| of F. The approximation algorithm then works like this. We define a sequence (mn )n≥1 of |Y |deg=1 such that: • ( mn )n≥1 is constant, • it is a Cauchy sequence, • lim vmn ( f ) = +∞. n→+∞ Write f = k≥0 [xk ]π k . 2 k is the same as the Newton polygon of g(T ) = k≥0 x k T ∈ O F T .

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