Lattices of Minimal Covolume in SL 2 : A Nonarchimedean Analogue of Siegel's Theorem μ≥π/21

Abstract

A classical result of Siegel [Si] asserts that if G = SL2(R) and ,u is its Haar measure (appropriately normalized) then mintj,u(F\G) = 7r/21 , where r runs over all the discrete subgroups of G. The minimum is obtained for the triangle group (2, 3, 7) which is a cocompact lattice in G (see also [Gr]). For G = SL2(C), Meyerhoff [Me] has shown recently that among the nonuniform lattices in G (that is, discrete subgroups of finite covolume but not cocompact) the minimum volume is obtained with F = SL2(&3) where &3 is the ring of integers of Q(v/3), and it is approximately 0.0863. (But, there are cocompact lattices with smaller covolumes and the minimum is not known, see [EGM, Th]). In this paper, we address the analogue question for locally compact nonarchimedean fields. Our main result is: Let F be a locally compact field of characteristic p > 0 with residue field of order q = pa . (Hence F is isomorphic to Fq((I/t)), the field of Laurent formal power series in l/t ).