M/Raum 515

According to a result of Conway, Rains and Sloane, similar sublattices of the root lattice A_4 are possible 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 quaternion algebra H (Q (/sqrt{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.

Prof. Dr. Michael Baake

Bielefeld