By Michael Harris (auth.), Yuri Tschinkel, Yuri Zarhin (eds.)
Algebra, mathematics, and Geometry: In Honor of Yu. I. Manin includes invited expository and study articles on new advancements bobbing up from Manin’s extraordinary contributions to arithmetic.
Contributors within the moment quantity: M. Harris, D. Kaledin, M. Kapranov, N.M. Katz, R.M. Kaufmann, J. Kollár, M. Kontsevich, M. Larsen, M. Markl, L. Merel, S.A. Merkulov, M.V. Movshev, E. Mukhin, J. Nekovár, V.V. Nikulin, O. Ogievetsky, F. Oort, D. Orlov, A. Panchishkin, I. Penkov, A. Polishchuk, P. Sarnak, V. Schechtman, V. Tarasov, A.S. Tikhomirov, J. Tsimerman, okay. Vankov, A. Varchenko, A. Vishik, A.A. Voronov, Yu. Zarhin, Th. Zink.
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Extra info for Algebra, Arithmetic, and Geometry: Volume II: In Honor of Yu. I. Manin
K) ∼ = H q Δopp , EnΔ ⊗ H q(Λ[n] , k). 1, H q Δopp , EnΔ ∼ = HH q(A, En ) ∼ = A⊗n . k , so that the second multiple H q(Λ[n] , k) is just k in degree 0. 9 is the following: the cyclic object A# associated to an algebra A inconveniently contains two things at the same time: the cyclic structure, which seems to be essential to the problem, and the bar resolution, which is needed only to compute the Hochschild homolq ogy HH q(A). Replacing A# with the cyclic complex L tr# A# ∈ D(Λ, k) disentangles these two.
And again, the same works for polylinear functors. In particular, given our k-linear abelian tensor category C, we can form the category Shv(C)# of pairs E, [n] , [n] ∈ Λ, E ∈ Shv(C n ), with a map from E , [n ] to E, [n] given by a pair of a map f : [n ] → [n] and either a map E → (f! )∗ E, or map (f! )! E → E; this is equivalent by adjunction. Then Shv(C)# is a biﬁbered category over Λ in the sense of [Gr]. The category of sections Λ → Shv(C)# of this biﬁbration can also be opp described as the full subcategory Shv(C# ) ⊂ Fun(C# , k) spanned by those opp opp opp n functors E# : C# → k-V ect whose restriction to (C ) ⊂ C# is a sheaf, that is, an object in Shv(C n ) ⊂ Fun((C opp )n , k).
Define ri , rj , rτ and rτ as above. Then there is a totally real Galois extension F ,+ /F + , linearly disjoint from M over F + , with the property that for i = 1, . . , m, (resp. j = 1, . . , m ) ri,F ,+ = ri |Γ ,+ (resp. rj,F ,+ ) corresponds to an automorphic representation F Πi of GL(2i, F ,+ ) ((resp. Πj of GL(2j, F ,+ )) and such that rτ (resp. rτ ) corresponds to an automorphic representation Πτ of GL(2m, F ,+ ) (resp. Πτ of GL(2m , F ,+ ). If F /F ,+ is a CM quadratic extension, then the base change Πi,F (resp.
Algebra, Arithmetic, and Geometry: Volume II: In Honor of Yu. I. Manin by Michael Harris (auth.), Yuri Tschinkel, Yuri Zarhin (eds.)