Class group of the group ring of a cyclic group of order p2 over the ring of integers in a quadratic number field

Abstract

Let Corn be the group ring of a cyclic group r,, of order p”+ ’ (p an odd prime) over the ring of integers 0 in a quadratic number field k (unramilied at p). Further let cl(cOr,,) denote the locally free class group of Or,. It is known that this group is mapped surjectively onto the class group cl(m) of the maximal order. cl(m) is the product of the ideal class groups of the fields occuring in the decomposition of kT, and we regard them as known. One is thus interested in the kernel group D(OT,,) of the map cl(OT) + cl(m), and it is this group for II = 1 that concerns us here. Using Cartesian diagrams and Mayer-Vietoris exact sequences for class groups we reduce the calculations of D(LoY,) to a number theoretic problem involving certain unit groups. We show that ID(Qr,)[ has a large p part which is minimal when p does not divide the class number h of k(<,), co being a primitive pth root of unity. In this connection we prove that if (p, h) = 1 then ID(6Jri)l =O(Or,)l, where for k real, G? is v~,~P~-‘, and for k imaginary, Q is vkPpP3. The p prime part of v~,~ is commensurate with IS(Or,)l and its p ‘part is at most p.