By Gérard Laumon, Jean Loup Waldspurger

Cohomology of Drinfeld Modular types presents an creation, in volumes, either to this topic and to the Langlands correspondence for functionality fields. it really is according to classes given by way of the writer who, to maintain the presentation as obtainable as attainable, considers the easier case of functionality instead of quantity fields; however, many very important gains can nonetheless be illustrated. a number of appendices on history fabric make this a self-contained e-book. will probably be welcomed via staff in algebraic quantity concept and illustration concept.

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PROOF. 17. 20. For the remainder of this section X denotes a uniform space. 21. REMARK. The following statements are pairwise equivalent: (1) T is weakly almost periodic; that is to say, if a is an index of X, then there exist a left syndetic subset A of T and a compact subset C of T such that x E X implies the existence of a subset B of T for which A C BC and xB Cxa. (2) If a is an index of X, then there exists a compact subset K of T such that x E X implies the existence of a subset A of T for which T = AK and xA C xa.

Is recursive at x. , is recursive at x, then S is recursive at x. Suppose Sx is recursive at x. Let U be an open neighborhood of x. , C SM- Let V be a neighborhood of x for which VM C U. , such that xA C V. Now xAM C U. Define B = S n AM. Since A C BM-\ B is a T-admissible subset of T. Also B C Sand xB C U. Thus S is recursive at x. It now follows that if T is recursive at x, then S is recursive at x. The converse is obvious. The proof is completed. 37. DEFINITION. Let T be a topological group.

The expression weakly almost periodic was introduced by Gottschalk [6]. 36) Cf. Gottschalk [2, 6, 8], Erdos and Stone [1], Gottschalk and Hedlund [5]. 38) The terms replete and extensive, as defined here, were introduced by Gottschalk and Hedlund [10]. If T is either 9 or (ft, a subset A of T is extensive if and only if A contains a sequence marching to + CD and a sequence marching to - CD. 55] The expression almost periodic, as applied to a point, is a generalization of the term recurrent as used by G.