The coordinator polynomial of some cyclotomic lattices

The coordinator polynomial of a lattice L is the numerator of its growth series as an abelian group, w.r.t. to a given set of generators S. We investigate the special case when L is the ring of integers of the cyclotomic field of order m and S is the corresponding set of unit roots. We compute it explicitly when m=p and m=2p, with p an odd prime. This confirms, for small p, a conjecture of Parker. Our approach is geometric and is grounded in the theory of Ehrhart polynomials.