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On the motive of a reductive group. (English) Zbl 0904.11014

Following Steinberg, a motive \(M\) of Artin-Tate type is attached to any connected reductive group \(G\) over a field \(k.\) When \(k\) is a local field of characteristic 0, the \(L\)-function \(L(M)\) is finite if and only if Serre’s Euler-Poincaré measure of \(G(k)\) is non-zero. In this case, a local function equation is obtained. When \(k\) is a number field, \(L(M)\) is used to evaluate certain adélic integrals in the trace formula.

MSC:

11G40 \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture
14A20 Generalizations (algebraic spaces, stacks)
11G09 Drinfel’d modules; higher-dimensional motives, etc.
20G30 Linear algebraic groups over global fields and their integers
20G35 Linear algebraic groups over adèles and other rings and schemes
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