By Gerard Laumon

Cohomology of Drinfeld Modular kinds goals to supply an creation to either the topic of the identify and the Langlands correspondence for functionality fields. those kinds are the analogs for functionality fields of Shimura forms over quantity fields. This current quantity is dedicated to the geometry of those forms and to the neighborhood harmonic research had to compute their cohomology. to maintain the presentation as obtainable as attainable, the writer considers the easier case of functionality instead of quantity fields; however, many vital gains can nonetheless be illustrated. it is going to be welcomed through employees in quantity idea and illustration concept.

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Xk , t | xt1 = φ(x1 ), . . , xtk = φ(xk ) for some injective endomorphism φ of Fk . For example, HT = x, y, t | txt−1 = xy, tyt−1 = yx , so the endomorphism φ is given by φ : x → xy, y → yx. Sapir: Residual properties of 1-relator groups 334 It is easy to see that every element in G has the form tk w(x1 , . . , xk )tl . If k +l = 0, then the element survives the homomorphism xi → 0, t → 1, G → Z. Hence we can assume that k + l = 0, and the element is conjugate to w(x1 , . . , xk ). Thus it is enough to consider elements from Fk only.