Lifting properties of prime geodesics (Q1024943)

The author continues the study begun by Peter Sarnak and Jeffrey Stopple of prime geodesics on \(\Gamma\backslash H\), \(\Gamma\) the modular group. Now \(\Gamma\) is allowed to be a Fuchsian group whose matrix entries lie in the ring of integers \(O_K\) of a number field \(K\). There is a one-to-one correspondence between the prime geodesics \(P\) on \(\Gamma\backslash H\) and the primitive hyperbolic conjugacy classes \(\{\gamma\}\) in \(\Gamma\). An eigenvalue \(\varepsilon\) of an element of \(\{\gamma\}\) determines a quadratic extension field \(K(\varepsilon)\) of \(K\). On the other hand, a prime ideal \(Q\) of \(O_K\) determines covering surfaces of \(\Gamma\backslash H\). A Frobenius map relates the lifting of \(P\) to the splitting of \(Q\) in \(K(\varepsilon)\).