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On the invariant \(M(A_{/K}, n)\) of Chen-Kuan for Galois representations. (English) Zbl 1310.11063

Summary: Let \(X\) be a finite set with a continuous action of the absolute Galois group of a global field \(K\). We suppose that \(X\) is unramified outside a finite set \(S\) of places of \(K\). For a place \(\mathfrak{p} \notin S\), let \(N_{X, \mathfrak{p}}\) be the number of fixed points of \(X\) by the Frobenius element \(\mathrm{Frob}_{\mathfrak{p}} \subset G_{K}\). We define the average value \(M(X)\) of \(N_{X, \mathfrak{p}}\) where \(\mathfrak{p}\) runs through the non-archimedean places in \(K\). This generalize the invariant of Chen-Kuan and we apply this for Galois representations. Our results show that there is a certain relationship between \(M(X)\) and the size of the image of Galois representations.

MSC:

11F80 Galois representations
11G05 Elliptic curves over global fields
11N45 Asymptotic results on counting functions for algebraic and topological structures
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