## Proceedings of the Japan Academy, Series A, Mathematical Sciences

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

Hyunsuk Moon

#### 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.

#### Article information

Source
Proc. Japan Acad. Ser. A Math. Sci., Volume 90, Number 7 (2014), 98-100.

Dates
First available in Project Euclid: 7 August 2014

https://projecteuclid.org/euclid.pja/1407415932

Digital Object Identifier
doi:10.3792/pjaa.90.98

Mathematical Reviews number (MathSciNet)
MR3249832

Zentralblatt MATH identifier
1310.11063

#### Citation

Moon, Hyunsuk. On the invariant $M(A_{/K}, n)$ of Chen-Kuan for Galois representations. Proc. Japan Acad. Ser. A Math. Sci. 90 (2014), no. 7, 98--100. doi:10.3792/pjaa.90.98. https://projecteuclid.org/euclid.pja/1407415932