The mass of unimodular lattices

Abstract

We derive an explicit formula for the mass of a unimodular Z-lattice of arbitrary signature using Bruhat-Tits theory.

Computation Of Some Zamolodchikov Volumes, With An Application

“…where the factor of R 3 represents levels of three hyperkähler moment maps and the S 1 is a period of a B-field. 4 Let v tr X (M ) denote the volume of the moduli space for S M X in the homogeneous metric induced from ds 2 = Tr D (E −1 dE) 2 . According to equation (2.20) the Zamolodchikov metric is related by rescaling with the factor 1/(2π 2 ) and so volumes are rescaled by 1/(π √ 2) D where D is the dimension of the moduli space.…”

“…Siegel's formulae involve products of "local density factors" over all prime numbers, and the computation of those densities is itself nontrivial. In the papers [4,14] the relevant factors were computed for the general case of unimodular lattices (not necessarily even). The paper [4] gives the volume using the measure µ = µ cpt normalized so that its analytic continuation to the connected compact group SO(D) gives unit total volume:…”

Finiteness theorems for congruence reflection groups

“…The list of the admissible fields is quite large here and we will not try the second step which involves a precise computation of the class numbers, this is more accessible in higher dimensions as we are going to see later on. r = 3 : We have (8). So d = 2 and using the additional information about class numbers we come the list of 4 real quadratic fields from part (v).…”

“…The following proposition presents some results of this kind. The reader can find more information on covolumes of arithmetic lattices defined by quadratic forms in [8] and [20].…”

Even unimodular Lorentzian lattices and hyperbolic volume

“…In §3-4 we discuss some interesting consequences of our main result. Finally, in §5 we discuss the case of Formulas (2)- (3). In particular, we state in Proposition 4 the exact relation between ∆ n and O(I n,1 ) when n ≡ 5 mod 8.…”