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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 $$Y\otimes_ k k_ r$$.

MSC:
 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