Volumes of symmetric spaces via lattice points

On volumes of quasi-arithmetic hyperbolic lattices

We prove that the covolume of any quasi-arithmetic hyperbolic lattice (a notion that generalizes the definition of arithmetic subgroups) is a rational multiple of the covolume of an arithmetic subgroup. As a corollary, we obtain a good description for the shape of the volumes of most of the known hyperbolic n-manifolds with n>3.

Arithmetic volumes of indefinite ternary lattices with square free discriminant

The main purpose of this paper is to give an explicit formula for the volumes of indefinite ternary lattices with square free discriminant. We also consider several examples of lattices, for which the fundamental domain is well-known and make sure the results obtained by our computation coincide with the known ones.

A volume estimate for the set of stable lattices

We show that in high dimensions the set of stable lattices is almost of full measure in the space of unimodular lattices.

Nonfreeness of algebras of symmetric Hilbert modular forms of even weight for $\mathbb{Q}(\sqrt{d})$ where d>5

We study the algebras of symmetric Hilbert modular forms of even weight for $\mathbb{Q}(\sqrt{d})$, considering them as modular forms for the orthogonal group of the lattice with signature (2,2). Comparing the volume of the corresponding symmetric domain with the volume of the Jacobian of the generators of these algebras, we prove that for all d, except for d=2, 3, 5 these algebras can't be free.

Volume and lattice points of reflexive simplices

We prove sharp upper bounds on the volume and the number of lattice points on edges of higher-dimensional reflexive simplices. These convex-geometric results are derived from new number-theoretic bounds on the denominators of unit fractions summing up to one. The main algebro-geometric application is a sharp upper bound on the anticanonical degree of higher-dimensional Q-factorial Gorenstein toric Fano varieties with Picard number one, where we completely characterize the case of equality.

格多边形的体积多项式

The volume polynomial of lattice polygons

本文证明每个具有非负整数系数的不定二次型均为一对格多边形的体积多项式，从而解决了平面上两个体的 Heine-Shephard 离散问题，并应用于构造具有给定相交数和自相交数的平面 tropical 曲线或环面曲面除子对。