Arithmetic Groups Research Articles

Finite index subgroups in Chevalley groups are bounded: An addendum to “On Bi-Invariant Word Metrics”

We prove that finite index subgroups in [Formula: see text]-arithmetic Chevalley groups are bounded.

On Galois groups of linearized polynomials related to the general linear group of prime degree

Let L(x) be any q-linearized polynomial with coefficients in Fq, of degree qn. We consider the Galois group of L(x)+tx over Fq(t), where t is transcendental over Fq. We prove that when n is a prime, the Galois group is always GL(n,q), except when L(x)=xqn. Equivalently, we prove that the arithmetic monodromy group of L(x)/x is GL(n,q), except when L(x)=xqn, and also equivalently, we prove that the image of the mod-(t) Galois representation of the Drinfeld module arising from L(x) is all of GL(n,q).

On the observability of Galois representations and the Tate conjecture

The Tate conjecture has two parts: i) Tate classes are linear combination of algebraic classes, ii) semisimplicity of Galois representations (for smooth projective varieties). B. Moonen proved that i) implies ii) in characteristic 0, using p-adic Hodge theory.We show that an unconditional result lies behind this implication: the observability of arithmetic monodromy groups of geometric origin (in any characteristic) - which leads to a sharpening of Moonen's result.We also discuss another aspect of the Tate conjecture related to the transcendence of p-adic periods.

Charmenability of higher rank arithmetic groups

Charmenability of higher rank arithmetic groups

The classification of 2-reflective modular forms

The classification of reflective modular forms is an important problem in the theory of automorphic forms on orthogonal groups. In this paper, we develop an approach based on the theory of Jacobi forms to give a full classification of 2-reflective modular forms. We prove that there are only 3 lattices of signature (2, n) having 2-reflective modular forms when n ≥ 14. We show that there are exactly 51 lattices of type 2U ⊕ L(−1) which admit 2-reflective modular forms and satisfy that L has 2-roots. We further determine all 2-reflective modular forms giving arithmetic hyperbolic 2-reflection groups. This is the first attempt to classify reflective modular forms on lattices of arbitrary level.

Arithmetic duality for two-dimensional local rings with perfect residue field

We give a refinement of Saito's arithmetic duality for two-dimensional local rings by giving algebraic group structures for arithmetic cohomology groups.

Applications of the Fixed Point Theorem for group actions on buildings to algebraic groups over polynomial rings

We apply the Fixed Point Theorem for the actions of finite groups on Bruhat–Tits buildings and their products to establish two results concerning the groups of points of reductive algebraic groups over polynomial rings in one variable, assuming that the base field is of characteristic zero. First, we prove that for a reductive k-group G, every finite subgroup of G(k[t]) is conjugate to a subgroup of G(k). This, in particular, implies that if k is a finite extension of the p-adic field Qp, then the group G(k[t]) has finitely many conjugacy classes of finite subgroups, which is a well-known property for arithmetic groups. Second, we give a short proof of the theorem of Raghunathan–Ramanathan [26] about G-torsors over the affine line.

A new proof of finiteness of maximal arithmetic reflection groups

A new proof of finiteness of maximal arithmetic reflection groups

Conjugation-invariant norms on SL2(R) for rings of S-algebraic integers with infinitely many units

This paper studies conjugation-invariant norms on for R a ring of S-algebraic integers with infinitely many units in a number field. We will show that barring some differences in the proofs the properties of these norms strongly resemble related properties of conjugation-invariant norms on arithmetic Chevalley groups of higher rank. Specifically, we prove two properties: First, we show that the diameter of each word norm on defined by finitely many conjugacy classes has an upper bound linear in the number of conjugacy-classes. Second, we show that all non-discrete conjugation-invariant norms on have a profinite norm-completion. Both of these properties of conjugation-invariant norms are already known for higher rank arithmetic Chevalley groups.

Aritmethic lattices of SO(1,n) and units of group rings

We establish that standard arithmetic subgroups of a special orthogonal group SO(1,n) are conjugacy separable. As an application we deduce this property for unit groups of certain integer group rings. We also prove that finite quotients of group of units of any of these group rings determines the original group ring.

Finiteness properties of automorphism spaces of manifolds with finite fundamental group

Given a closed smooth manifold M of even dimension 2n≥6\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$2n\\ge 6$$\\end{document} with finite fundamental group, we show that the classifying space BDiff(M)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$B\ extrm{Diff}(M)$$\\end{document} of the diffeomorphism group of M is of finite type and has finitely generated homotopy groups in every degree. We also prove a variant of this result for manifolds with boundary and deduce that the space of smooth embeddings of a compact submanifold N⊂M\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$N\\subset M$$\\end{document} of arbitrary codimension into M has finitely generated higher homotopy groups based at the inclusion, provided the fundamental group of the complement is finite. As an intermediate result, we show that the group of homotopy classes of simple homotopy self-equivalences of a finite CW complex with finite fundamental group is up to finite kernel commensurable to an arithmetic group.

On superintegral Kleinian sphere packings, bugs, and arithmetic groups

Abstract We develop the notion of a Kleinian Sphere Packing, a generalization of “crystallographic” (Apollonian-like) sphere packings defined in [A. Kontorovich and K. Nakamura, Geometry and arithmetic of crystallographic sphere packings, Proc. Natl. Acad. Sci. USA 116 2019, 2, 436–441]. Unlike crystallographic packings, Kleinian packings exist in all dimensions, as do “superintegral” such. We extend the Arithmeticity Theorem to Kleinian packings, that is, the superintegral ones come from ℚ {{\mathbb{Q}}} -arithmetic lattices of simplest type. The same holds for more general objects we call Kleinian Bugs, in which the spheres need not be disjoint but can meet with dihedral angles π m {\frac{\pi}{m}} for finitely many m. We settle two questions from Kontorovich and Nakamura (2019): (i) that the Arithmeticity Theorem is in general false over number fields, and (ii) that integral packings only arise from non-uniform lattices.

A genus two arithmetic Siegel-Weil formula on 𝑋₀(𝑁)

We define a family of arithmetic zero cycles in the arithmetic Chow group of a modular curve X 0 ( N ) X_0(N) , for N > 3 N>3 odd and squarefree, and identify the arithmetic degrees of these cycles as q q -coefficients of the central derivative of a Siegel Eisenstein series of genus two. This parallels work of Kudla-Rapoport-Yang for Shimura curves.

Norm rigidity for arithmetic and profinite groups

AbstractLet be a commutative ring, and assume that every non‐trivial ideal of has finite index. We show that if has bounded elementary generation then every conjugation‐invariant norm on it is either discrete or precompact. If is any group satisfying this dichotomy, we say that has the dichotomy property. We relate the dichotomy property, as well as some natural variants of it, to other rigidity results in the theory of arithmetic and profinite groups such as the celebrated normal subgroup theorem of Margulis and the seminal work of Nikolov and Segal. As a consequence we derive constraints to the possible approximations of certain non‐residually finite central extensions of arithmetic groups, which we hope might have further applications in the study of sofic groups. In the last section we provide several open problems for further research.

Irregular cusps of ball quotients

AbstractWe study the branch divisors on the boundary of the canonical toroidal compactification of ball quotients. We show a criterion, the low slope cusp form trick, for proving that ball quotients are of general type. Moreover, we classify when irregular cusps exist in the case of the discriminant kernel and construct concrete examples for some arithmetic subgroups. As another direction of study, when a complex ball is embedded into a Hermitian symmetric domain of type IV, we determine when regular or irregular cusps map to regular or irregular cusps studied by Ma.

Effective bounds for Vinberg’s algorithm for arithmetic hyperbolic lattices

A group of isometries of a hyperbolic n-space is called a reflection group if it is generated by reflections in hyperbolic hyperplanes. Vinberg gave a semi-algorithm for finding a maximal reflection sublattice in a given arithmetic subgroup of $${{\,\textrm{O}\,}}(n,1)$$ of the simplest type. We provide an effective termination condition for Vinberg’s semi-algorithm with which it becomes an algorithm for finding maximal reflection sublattices. The main new ingredient of the proof is an upper bound for the number of faces of an arithmetic hyperbolic Coxeter polyhedron in terms of its volume.

Volumes of spheres and special values of zeta functions of $\mathbb{Z}$ and $\mathbb{Z}/n\mathbb{Z}$

The volume of the unit sphere in every dimension is given an interpretation as a product of special values of the zeta function of $\mathbb {Z}$, akin to volume formulas of Minkowski and Siegel in the theory of arithmetic groups. A product formula is foun

Abelianization and fixed point properties of units in integral group rings

AbstractLet G be a finite group and the unit group of the integral group ring . We prove a unit theorem, namely, a characterization of when satisfies Kazhdan's property , both in terms of the finite group G and in terms of the simple components of the semisimple algebra . Furthermore, it is shown that for , this property is equivalent to the weaker property (i.e., every subgroup of finite index has finite abelianization), and in particular also to a hereditary version of Serre's property , denoted . More precisely, it is described when all subgroups of finite index in have both finite abelianization and are not a nontrivial amalgamated product. A crucial step for this is a reduction to arithmetic groups , where is an order in a finite‐dimensional semisimple ‐algebra D, and finite groups G, which have the so‐called cut property. For such groups G, we describe the simple epimorphic images of . The proof of the unit theorem fundamentally relies on fixed point properties and the abelianization of the elementary subgroups of . These groups are well understood except in the degenerate case of lower rank, that is, for with an order in a division algebra D with a finite number of units. In this setting, we determine Serre's property FA for and its subgroups of finite index. We construct a generic and computable exact sequence describing its abelianization, affording a closed formula for its ‐rank.

On Farrell–Tate cohomology of GL3 over rings of quadratic integers

The goal of the present paper is to push forward the frontiers of computations on mod ℓ Farrell–Tate cohomology for arithmetic groups. We deal with ℓ-rank 1 cases different from PSL2. The conjugacy classification of cyclic subgroups of order ℓ is reduced to the classification of modules of Cℓ-group rings over suitable rings of integers which are principal ideal domains, generalizing an old result of Reiner. As an example of the number-theoretic input required for the Farrell–Tate cohomology computations, we discuss in detail the homological torsion in PGL3 over principal ideal rings of quadratic integers, accompanied by machine computations in the imaginary quadratic case.

Square integrals of the logarithmic derivatives of Selberg’s zeta functions in the critical strip

In our previous work (Y. Hashimoto, Selberg’s zeta function for the modular group in the critical strip, Math. Nachr. 294 (2021) 1899–1904, https://doi.org/10.1002/mana.202000268 ), we proposed an upper bound of the logarithmic derivative of Selberg’s zeta function for the modular group in the critical strip. This paper studies the growth of its square integral for the modular group, co-compact arithmetic groups derived from indefinite quaternion algebras and their subgroups.