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X,~ (resp. gfdx,~,~ ). We have a forgetful morphism of stacks ~'EX ~ (resp. ~'{x,~, r~ I .... ' ~ffffX,-@, I ) 256 G. Laumon et al. which maps (E, (),, ~)) into E and a morphism of stacks ~ - ~ x ,~ (resp. 2 , Spec(~:(~)~) gYgx,~, t , Spec(x(m)~)) which maps (E, (2, ~)) into 2 and which lifts i~,o ~ r~ where io~,o is the pole. It is clear that r~,~ is the base change ofro~ by the canonical morphism of stacks Therefore, if I ~ J c X \ { o o } are two finite closed subschemes, we have a 2-commutative diagram of stacks g f f x,~, ~ > , g f f x,~ with 2-cartesian squares.

H. (K . . c~(g . oo,o ) - 1K . . g. ), h]GLa_h(Oo), m~] . is in Fix~(0~, g )~e,n)if and only if there exists 6eA • k ~ e K ~ , k ~ , ~ k~eGLa_h((9o) such that h~, o = 6hO~,o(g~,o)-lk~,O h~. = 6h~ok~o m~ + r = (deg(O)/deg(o))6(rn(6)) + m~ ~,~ and 278 G. Laumon et al. e. if and only if there exists 6cA • such that ~](h ~,~ - a6h~,O e K ~,O (l~,o (h~)- 16h~eGLd_h(Co) I Qdeg(~)6(rn(6)) = r deg(o) . 3) Lemma. For any h~'~176176 • h~GLd-h(F~), the only 6~A • such that f (h ~ ~,O ~ K ~,O ~(h~o) - 16h~oeGLd_h((9o) { ~(rn(6)) = 0 is the identity element (recall that K ~'~ is small enough).

Let us set 21 = 2 o F r o b s . F r o m = we get an isomorphism of Co~(~w, (gs-modules ~ :@d,t(io~,o)(~Ma,~(ioo,o)) ~ > ~'(~/1'o) = ~1'_ 1 which commutes again with the ~b's. o)(~Md, i(i| which commutes with the ~,'s. i(1)) = ei+l,j(0)(Vi = 0 . . . d e g ( ~ ) - 2, Vj = 1 . . . 2)). So we can set a t = ~1 ~ c a n . Then we let l~7Z act on ~f::x,~ (resp. , (21, ~ ) ) . We let the reader check that this action of 7 / o n ~-~x,~ (resp. 8 ~ x , e , ~ ) commutes with the action of Do~/wo~. -• z As this action of Z on the stacks ~ : : x , ~ , t ' s commutes with the transition maps fx and rj,~, we get for any finite set of places oo ~ T c ]X I an action o f T / o n g'f':~:,~ which lifts the action of 2g on dY:~r and which commutes with the action of (/3o~/m z ) x (Dr) • .

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