Zur Frobeniusoperation auf der Homologie einiger arithmetischer Gruppen. (On the Frobenius operation on the homology of some arithmetic groups) (Q1101826)

Let \({\mathfrak a}\) be an ideal of the polynomial ring \({\mathbb{F}}_ q[t]\), \(\Gamma_{{\mathfrak a}}=\{\gamma \in GL_ 2({\mathbb{F}}_ q[t])|\gamma\equiv 1(mod {\mathfrak a})\}^ a \)congruence subgroup. The Frobenius endomorphism Frob: \(\Gamma\) \({}_{{\mathfrak a}}\to \Gamma_{{\mathfrak a}}\), \(\left( \begin{matrix} a\quad b\\ c\quad d\end{matrix} \right)\mapsto \left( \begin{matrix} a\quad p b p\\ c\quad p d p\end{matrix} \right)\), induces a linear transformation \((Frob)_*: H_ 1(\Gamma_{{\mathfrak a}},{\mathbb{Q}})\to H_ 1(\Gamma_{{\mathfrak a}},{\mathbb{Q}})\). In this paper it is shown that the eigenvalues of \((Frob)_*\) are roots of unity or zero. A formula for calculating the trace of \((Frob)^{{\mathfrak a}}_*\) is given as well.