By Gérard Laumon, Jean Loup Waldspurger

Cohomology of Drinfeld Modular forms presents an creation, in volumes, either to this topic and to the Langlands correspondence for functionality fields. it truly is in accordance with classes given by means of the writer who, to maintain the presentation as obtainable as attainable, considers the better case of functionality instead of quantity fields; however, many vital positive aspects can nonetheless be illustrated. a number of appendices on heritage fabric make this a self-contained e-book. it is going to be welcomed through staff in algebraic quantity idea and illustration concept.

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Let I be a regular element in T. 1, there exists 1' E H which is stably conjugate but not conjugate to I in G. (T, G) = Z/2. 8. Singular semisimple elements. Let G = U(3). For ~ E F*, let He be the unitary group in two variables defined by the Hermitian form: He The isomorphism class of depends only on ~ modulo NE* and we obtain a bijection between F* /NE* and the set of isomorphism classes of unitary groups in two variables over F with respect to E. The group H 1 is quasisplit and is isomorphic to U(2).

Let IP(T) be the set of unramified characters of T. (T, G)-orbit and every element of E"( G) is of the form 7r x for some x E Il"(T). Fix an element Wp E Wp whose projection to r(Fun /F) is the Frobenius element. (T, G) (and hence E"(G)) and the set of semisimple G- conjugacy classes in LG of the form {g x wp }. The conjugacy class {g(7r)} in LG associated to a representation 7r E E" ( G) is called the Langlands class of 7r. We can choose a representative g x wp with g E f. Let (H, s, T/) be an endoscopic datum for G.

Regular class in G. Suppose that 1' E TH. Then 1' is called ( G, H)-regular if a('ef;(I')) f. 1 for each root a of T which is not the image of a root of TH in H. Let 1' be a ( G, H)-regular element of TH. Suppose that 'lj; is defined over F (this entails no loss of generality since the choice of Tis arbitrary) and let I= 'ef;(1'). H in H -Y, and that of T in G-y. It follows that 'lj; extends to an isomorphism of H-y' with G-Y which is an inner twisting over F. In particular, we can identify Z(H-y') with Z(G-y) and, if Fis local, we can choose compatible measures on H-y' and G-y.