On torsion in the cohomology of certain \(S\)-arithmetic groups
As in the case of arithmetic groups there is a close relation between the cohomology of \(S\)-arithmetic groups and the theory of automorphic forms. Congruences for modular forms or \(p\)-adic properties of special values of \(L\)-functions can be derived by using group cohomological methods. As parts of the general study of the cohomology of \(S\)-arithmetic groups it turned out to be of interest to detect torsion classes in the integral cohomology. This paper deals with the cohomology of the group \(\Gamma_ p= \text{SL}_ 2 (\mathbb{Z} [{1 \over p}])\) with respect to the coefficient system \(M_ n (R_ p)\) given by the module of homogeneous polynomials of even degree \(n\) in two variables over the ring \(R_ p= \mathbb{Z} [{1 \over p}]\), \(p\) a prime. The main result says that the torsion elements in \(H^* (\Gamma_ p, M_ n (R_ p))\) which are prime to \(6_ p\) do not depend on \(p\). As applications some congruences between newforms for \(\Gamma_ 0 (p)\) and \(\Gamma_ 0 (q)\), \(p\), \(q\) primes, are given. This paper is an abbreviated version of the author's thesis [Kohomologie spezieller \(S\)-arithmetischer Gruppen und Modulformen, Bonner Math. Schriften 264 (1994; Zbl 0819.11020)]reviewed below.
scientific article; zbMATH DE number 599666
scientific article