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Effective versions of the Chebotarev density theorem for function fields. (English. Abridged French version) Zbl 0822.11077
Let \(k\) be a finite field with algebraic closure \(\overline {k}\), let \(\varphi\in \text{Gal} (\overline {k}/k)\) be the Frobenius map \(x\mapsto x^{| k|}\) and denote by \(k_ r\) the extension of \(k\) of degree \(r\). Let \(X\to Y\) be a finite Galois covering of smooth projective curves defined over \(k\) with Galois group \(G\) and let \(k_ m\) be the algebraic closure of \(k\) in the function field \(k(X)\). Further let \(\pi(r)\) denote the number of unramified points of \(Y\) having degree \(r\) and \(\pi_ C(r)\) the number of those among them, whose Frobenius conjugacy class equals \(C\). The authors obtain explicit bounds for the difference \[ \Biggl| \pi_ C (r)- m{{| C|} \over {| G|}} \pi(r) \Biggr| \] for the conjugacy classes \(C\) whose restriction to \(k_ m\) equals \(\varphi^ r\). If the last condition is not satisfied, then \(\pi_ C(r) =0\). If \(m=1\), i.e. the covering is geometric, then an analogous result is obtained when in the definition of \(\pi\) and \(\pi_ C\) one replaces \(Y\) with \(Y\otimes_ k k_ r\).

11R58 Arithmetic theory of algebraic function fields
11R45 Density theorems
14H05 Algebraic functions and function fields in algebraic geometry
11G20 Curves over finite and local fields