PUB - Publikationen an der Universität Bielefeld

Similar sublattices of the root lattice A(4) are possible [J.H. Conway, E.M. Rains, N.J.A. Sloane, On the existence of similar sublattices, Can. J. Math. 51 (1999) 1300-1306] for each index that is the square of a non-zero integer of the form m(2) + mn - n(2). Here, we add a constructive approach, based on the arithmetic of the quaternion algebra H(Q(root 5)) and the existence of a particular involution of the second kind, which also provides the actual sublattices and the number of different solutions for a given index. The corresponding Dirichlet series generating function is closely related to the zeta function of the icosian ring. (c) 2008 Michael Baake. Published by Elsevier Inc. All rights reserved.