On the cuspidal cohomology of the Bianchi modular groups (Q790940)

Denote by \(d\) a square free negative integer, by \(\ell ={\mathbb{Q}}(\sqrt{d})\) the corresponding imaginary quadratic field and by \({\mathcal O}_ d\) the ring of integers in \(\ell\). Let \(h(d)\) be the class number of \({\mathcal O}_ d\). Let \(\Gamma:=SL_ 2({\mathcal O}_ d)/\{\pm 1\}\) act on \(X:=SU(2)\backslash SL_ 2({\mathbb{C}})\) and denote by \(H^*_{cusp}(X/\Gamma,{\mathbb{C}})\) the image of the cohomology with compact support \(H^*_ c(X/\Gamma,{\mathbb{C}})\) in \(H^*(X/\Gamma,{\mathbb{C}})\). As a main result the following estimate is obtained: \[ \dim H^ 1_{cusp}(X/\Gamma,{\mathbb{C}})\geq \phi(d)/24-1/4-\frac{1}{2}h(d). \] Here \(\phi(\;)\) is the Euler-phi-function. The result follows from the computation of the Lefschetz number of the involution \(\tau\) which is induced on \(X/\Gamma\) by the Galois action of \(\ell| {\mathbb{Q}}\). Using the above estimate, results of Grunewald and Schwermer and unpublished explicit computations of N. Krämer (Bonn) for small \(d\), a complete list of all \(d\)'s such that \(H^ 1_{cusp}(SL_ 2({\mathcal O}_ d)/\{\pm 1\},{\mathbb{C}})=\{0\}\) is obtained.