On the distribution of class groups of number fields

We propose a modification of the predictions of the Cohen--Lenstra heuristic for class groups of number fields in the case where roots of unity are present in the base field. As evidence for this modified formula we provide a large set of computational data which show close agreement.Comment: 14 pages. To appear in Experimental Mat

A Murnaghan--Nakayama rule for values of unipotent characters in classical groups

We derive a Murnaghan--Nakayama type formula for the values of unipotent characters of finite classical groups on regular semisimple elements. This relies on Asai's explicit decomposition of Lusztig restriction. We use our formula to show that most complex irreducible characters vanish on some $\ell$-singular element for certain primes $\ell$. As an application we classify the simple endotrivial modules of the finite quasi-simple classical groups. As a further application we show that for finite simple classical groups and primes $\ell\ge3$ the first Cartan invariant in the principal $\ell$-block is larger than~2 unless Sylow $\ell$-subgroups are cyclic.Comment: Added a missing assumption to the statement of theorem 2. 25.11.16: Added a corrigendum to the proof of Thm. 7.