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6)) and any finitely generated ~x-submodule of the free Dx-module of rank one, V~~, induces a ~-lattice M~ ~ V~ such that ~o~(M~) = M~. Finally, ifx = oo (resp. x = o), we have seen that ( v ~ , q ~ ) = (v;~, ~o~,)~ (resp. (Vo, q~o) = (Vo,, q~o) , d) as a module over D~ = IMa(F| (resp. Do = ~4Id(Fo)). 8) there exists a lattice M " c V " (resp. M'o ~ V'o) such that M" (r176 = ~ (M'~) = wg 1 M " dimk(qg~(M'~)/M'~) = 1 (resp. woM'o c (P'o(M'o) ~ M'o dimk (M'o/q" ( M ' ) ) = 1 and the support of M'o/q~'o (M'o) is the connected component of Spec(K(o) | k) c Spec(Co | k) which corresponds to the given embedding x(o) ~ k) 268 G.

We can factor r~ through where gfEx, e is viewed as a K(oo)-stack by the pole map. -• Let us begin with the representability. Let S be a scheme (over let E = (~i, j~, t~),~z be a ~-elliptic sheaf over S and let 2:S ~ Spec(x(~)d) lFq-morphism of schemes which lifts the pole i ~ , o : S ~ {~} of E. For each negative interger n, let J. be the f p p f sheaf , t ~,O)/~iI~ M e , 1 (ion,o), proIFq), be a non- ~,, o/rn oo ~rb) over S. ~ ''. +i ~ J. is the reduction modulo w~+~. 's are clearly representable by S-schemes and the transition maps are all affine and locally finitely presented.

2~ ~Doo/vJ~, - • z resp. K ~ ' ~ 1 7 6 on f~,o} is the action of right translations on the factor Y~ = 7/(resp. D~/~oo, - • z resp. (D~'~215 Moreover, the action of Frob~o on M(k)tr ' fz)is induced by the translation (r deg(o), r) on the factor Y~ x Y~ = 7/• 7l of ~'7~'~ the correspondence M(g~'~ Ol / M(k)(L ~t * - - - M(k)tr, nl is induced by the correspondence (D ~,o)• (D~,O)• ~,o n (g oo,o)- ~K ~,Og ~,o) <5 (D~,O)• ~,o where cl(h~'~ ~'~ c~ ( g ~ ' ~ 1 7 6 1 7 6 = h~'~ cz(h~'~ ~'~ c~ (g~176176176176 = h~'~ ~'~ on the factor (D~'~ x of fk~,o} and the action of ( -D• ~ / ~ )z/ K ~induced by the action by right translations of D~/vJ~o - • z on itself.