Gross, Benedict H. On the motive of a reductive group. (English) Zbl 0904.11014 Invent. Math. 130, No. 2, 287-313 (1997). 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. Reviewer: L.N.Vaserstein (University Park) Cited in 3 ReviewsCited in 46 Documents 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 Keywords:motive of Artin-Tate type; connected reductive group; \(L\)-function; number field × Cite Format Result Cite Review PDF Full Text: DOI