HIGHER CLASS GROUPS OF GENERALIZED EICHLER ORDERS

Abstract

ABSTRACT In this paper we study the possible torsion in even-dimensional higher class groups Cl 2n (Λ)(n ≥ 1) of an order Λ in a semisimple algebra A over a number field F with a ring of integers 𝒪 F . We show that for certain orders, called generalized Eichler orders, bip-torsion in Cl 2n (Λ) can only occur for primes p dividing prime ideals ℘ of 𝒪 F , at which Λ is not maximal. In particular, the results apply to Eichler orders in quaternion algebras and to hereditary orders.