For an arithmetic surface $Y$ and a finite group $G$; we can define the second adelic Chow group of locally free $O_Y[G]$-modules. This Chow group supports a natural map to the usual class group of locally free $\mathbb{Z}[G]$-modules. For a locally free $O_Y [G]$-module $M$, which satisfies certain conditions, we construct an adelic second Chern class $c_2 (M)$ in the adelic Chow group. We can then prove a Riemann Roch theorem which permits us to calculate the Euler characteristic of $M$.

The construction of the adelic second Chern class leads naturally to the study of $SK_1$ of certain rings associated to $O_Y [G]$. In the last part of the talk we will describe an exponential function which takes values in $SK_1$ of these rings.