${\rm SL}_{2}$ over complex quadratic number fields. I

In the present paper, we study some congruence subgroups of $PSL_2(\sigma)$ where $\sigma$ is the ring of integers in $k=Q(\sqrt{-d})$. For decomposed primes, and for $d=1,3$, there is a certain compact oriented closed topological $3$-manifold which occurs naturally. Its fundamental group is a quotient of the subgroup. We define an adjusted version of the Hecke algebra which is an algebra of endomorphisms of the commutator quotient group. There seems to exist many subgroups for which the commutator quotient group has rank one. For one such case, we exhibit an elliptic curve, defined over $k$, which seems to have the property that its Hasse–Weil $\zeta$-function coincides with the Dirichlet series arising from the Hecke algebra. In the last part, we show that one can adopt a topological method of H. Zimmert, and obtain estimates for congruence subgroups of inert primes.