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.


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