分野別セミナー

A tree-theoretic approach to classify lattices in PGL_2 of
p-adic fields, developed in my past works, partly by collaboration
with G.Cornelissen and A.Kontogeorgis, gave an effective way to
describe lattices in this p-adic Lie group, and was applied to
numerous problems in geometry of algebraic curves. Daniel Allcock and
I tried to carry out the similar story for PGL_3. What we obtained so
far are the following results, which I am going to speak about:
PGL_3(Q_2) has exactly two lattices of minimal covolume, which are
both arithmetic, commensurable to each other; moreover, one of these
two lattices is the one constructed by Mumford in his famous
construction of a fake projective plane.