On the invariant $M(A_{/K}, n)$ of Chen-Kuan for Galois representations

Abstract

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.