A motivic Chebotarev density theorem
Let \(E\) be a field of characteristic 0 containing the roots of unity and let \(C\) be an \(E\)-linear additive pseudo-abelian rigid tensor category. For \(X\in C\), the zeta function \(Z(X,t)\) is a certain formal power series in \(K_0(C)t\). Assuming a finite group \(G\) acts on \(X\), let \(L(t,X,\rho)=Z((V\otimes X)^G,t)\), where \(V\) is a finite-dimensional \(E\)-vector space and \(\rho:G\to\text{GL}(V)\) is a representation of \(G\). On specialising \(C\) to the category \({\mathcal M}_k(E)\) of Chow motives over a field \(k\) with coefficients in \(E\), the authors prove that their \(L\)-function \(L(t,X,\rho)\) is a rational function of \(t\) in this case. Moreover, if \(k\) is a finite field, then the function \(L(t,X,\rho)\) is proved to specialise to the usual Artin \(L\)-function under the trace of the Frobenius automorphism. Finally, the authors define the motive of Artin symbols and obtain an expression for that motive, which can be viewed as a Chebotarev density theorem for function fields, along the lines of the work: \textit{V. K. Murty} and \textit{J. Scherk} [C. R. Acad. Sci., Paris, Ser. I 319, No. 6, 523--528 (1994; Zbl 0822.11077)].
scientific article; zbMATH DE number 5074578