# Download e-book for iPad: On the Cohomology of Certain Non-Compact Shimura Varieties by Sophie Morel

By Sophie Morel

This booklet stories the intersection cohomology of the Shimura kinds linked to unitary teams of any rank over Q. more often than not, those forms are usually not compact. The intersection cohomology of the Shimura type linked to a reductive team G consists of commuting activities of absolutely the Galois crew of the reflex box and of the crowd G(Af) of finite adelic issues of G. the second one motion may be studied at the set of complicated issues of the Shimura sort. during this e-book, Sophie Morel identifies the Galois action--at reliable places--on the G(Af)-isotypical elements of the cohomology.

Morel makes use of the strategy constructed by means of Langlands, Ihara, and Kottwitz, that's to match the Grothendieck-Lefschetz mounted aspect formulation and the Arthur-Selberg hint formulation. the 1st challenge, that of making use of the fastened element formulation to the intersection cohomology, is geometric in nature and is the item of the 1st bankruptcy, which builds on Morel's earlier paintings. She then turns to the group-theoretical challenge of evaluating those effects with the hint formulation, while G is a unitary team over Q. functions are then given. specifically, the Galois illustration on a G(Af)-isotypical element of the cohomology is pointed out at just about all areas, modulo a non-explicit multiplicity. Morel additionally provides a few effects on base swap from unitary teams to normal linear groups.

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**Extra info for On the Cohomology of Certain Non-Compact Shimura Varieties**

**Example text**

Vol(K p ) p For every γ ∈ G(Af ), write Oγ (f p ) = p p f p (x −1 γ x)dx, G(Af )γ \G(Af ) p p where G(Af )γ is the centralizer of γ in G(Af ). Remember that we fixed an injection F ⊂ Qp ; this determines a place ℘ of F over p. Let Qnr p be the maximal unramified extension of Qp in Qp , L be the unramified extension of degree j of F℘ in Qp , r = [L : Qp ], L be a uniformizer of L and σ ∈ Gal(Qnr p /Qp ) be the element lifting the arithmetic Frobenius morphism of Gal(F/Fp ). Let δ ∈ G(L). Define the norm N δ of δ by N δ = δσ (δ) .

R1 + · · · + rm } and r = r1 + · · · + rm . Then there is an isomorphism MS ∼ −→ RE/Q GLr1 × · · · × RE/Q GLrm × GU(p − r, q − r) that sends diag(g1 , . . , gm , g, hm , . . , h1 ) to (c(g)−1 g1 , . . , c(g)−1 gm , g). The inverse image by this isomorphism of RE/Q GLr1 × · · · × RE/Q GLrm is called the linear part of MS and denoted by LS (or LPS ). The inverse image of GU(p − r, q − r) is called the hermitian part of MS and denoted by Gr (or GPS ). 1. 1. It is enough to consider the quasi-split forms.

R} − − and that n1 + · · · + nr is even. 1. The derived group of G is simply connected, so, by proposition 1 of [L2], there exists a L-morphism η : L H := H WQ −→ L G := G WQ extending η0 : H −→ G. We want to give an explicit formula for such a η. 1 an injection Q ⊂ Qv ; this gives a morphism Gal(Qv /Qv ) −→ Gal(Q/Q), and we fix a morphism WQv −→ WQ above this morphism of Galois groups. Let ωE/Q : A× /Q× −→ {±1} be the quadratic character of E/Q. 2 and is easy to prove. 2 Let µ : WE −→ C× be the character corresponding by the × class field isomorphism WEab A× E /E to a character extending ωE/Q .