Joint Colloquium: Baake -- Similar sublattices of the root lattice $A_4$

Among the sublattices of a given lattice in Euclidean space, those similar to
the original lattice form an interesting and important subclass. In recent
years, several attempts have been made to classify them, with limited success
so far. An exception are lattices in dimensions up to 4, and some of the root
lattices. The latter were studied by Conway, Rains and Sloane in a non-
constructive manner by means of quadratic forms, which gave access to the 
possible sublattice indices, but not to the sublattices themselves.

In particular, the root lattice $A_4$ could not be treated completely. 
It is the purpose of this talk to add a constructive approach, based on the 
arithmetic of a certain quaternion algebra and the existence of an unusual
involution of the second kind. This 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.

This is joint work with Manuela Heuer and Robert V. Moody.
Michael Baake (Bielfeld)

Note: the JC organizers apologize for the short notice of this  talk.