## D-elliptic sheaves and the langlands correspondence by Laumon G., Rapoport M., Stuhler U.

By Laumon G., Rapoport M., Stuhler U.

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Example text

0). The sequence becomes stationary with non-trivial end term. On the other hand, Im(O')/Im(O '+ 1) is a D | x'-module, hence its dimension as a vector space over ~' = F | ~cis divisible by d. Hence it follows that n < d - 1, and that putting N = Im(O"), dim A7 = rd, O

This action commutes with ~ and, for each ie 77, stabilizes _M~c/V. Thanks to our identification of N~ with IMa((9~), we get canonical splittings and Now, N-' is a locally free F~ ~ (gs-module of constant rank d and ~ c N ' is a locally free 0~ (~ (gs-submodule of constant rank d, for each i e 7/. cMi_ 1 ~M~ ~-elliptic sheaves and the Langlands correspondence 0'(~M'~) = * M i -, 1 253 , so that vr ~! T ~l and that the quotient K ( ~ ) | Cs-module Mi/vJ~M,--,,~ M~/~'(~M'i) (viewed as an (9~/• schemes is supported on the graph of a IFq-morphism of i~,,:s--, {~} and is locally free of constant rank one on its support, for each i 9 7/.

4), the map r~,o(k) : 8Efx, ~,o(k) ~ ~fEx, ~,o(k) can be described in the following way. e. /17/'~ --L)lT/~o be the restriction to 37/" of the F~ (~ frobq-semilinear map (q3~)-1 : V" ~ V~. Here we have split (Voo,~o~) andMoo into (V~, q~)d and ( M ' ) d using the identification ~ = ~/~d((9o~) and ~b~ : V~o ~ V~ is the dual map of ~o'. 6). Therefore, in terms of the description A • \Y~,0 of the isogeny class gg#x, ~,o(k)(~, r~), the restriction r~,o(k)(p, n): ~-~x, ~,o(k)(p, n) ~ g~f x, ~,o(k)ltr, ~) of r| to this isogeny class can be described in the following way.