BOUNDS FOR ARITHMETIC HYPERBOLIC REFLECTION GROUPS IN DIMENSION 2

In [1], Borel discussed discrete arithmetic groups arising from quaternion algebras over number fields with particular reference to arithmetic Kleinian and arithmetic Fuchsian groups. In these cases, he described, in each commensurability class, a class of groups which contains all maximal groups. Developing results on embedding commutative orders of the defining number field into maximal or Eichler orders in the defining quaternion algebra, Chinburg and Friedman [2] stated necessary and sufficient conditions for the existence of torsion in this class of groups in terms of the defining arithmetic data. This was more fully explored in the case of Kleinian groups in [3]. In the case of Fuchsian groups, these results on the existence of torsion were extended to obtain formulas for the number of conjugacy classes of finite cyclic subgroups for each group in this class [8, 9]. In this paper, we examine, across the range of arithmetic Fuchsian groups, how widespread torsion is in maximal Fuchsian groups. Some studies in low genus cases (see e.g. [7, 12]) indicate that 2-torsion is very prevalent. The results obtained here substantiate that but we will also obtain maximal arithmetic Fuchsian groups which are torsion-free. The author is grateful to Alan Reid for conversations on parts of this paper.

Arithmetic Fuchsian groups are the most interesting and most important Fuchsian groups owing to their significance for number theory and owing to their geometric properties. However, for a fixed signature there exist only finitely many non-conjugate arithmetic Fuchsian groups; it is therefore desirable to extend this class of Fuchsian groups. This is the motivation of our definition of semi-arithmetic Fuchsian groups. Such a group may be defined as follows (for the precise formulation see Section 2). Let Γ be a cofinite Fuchsian group and let Γ2 be the subgroup generated by the squares of the elements of Γ. Then Γ is semi-arithmetic if Γ is contained in an arithmetic group Δ acting on a product Hr of upper halfplanes. Equivalently, Γ is semi-arithmetic if all traces of elements of Γ2 are algebraic integers of a totally real field. Well-known examples of semi-arithmetic Fuchsian groups are the triangle groups (and their subgroups of finite index) which are almost all non-arithmetic with the exception of 85 triangle groups listed by Takeuchi [16]. While it is still an open question as to what extent the non-arithmetic Fuchsian triangle groups share the geometric properties of arithmetic groups, it is a fact that their automorphic forms share certain arithmetic properties with modular forms for arithmetic groups. This has been clarified by Cohen and Wolfart [5] who proved that every Fuchsian triangle group Γ admits a modular embedding, meaning that there exists an arithmetic group Δ acting on Hr, a natural group inclusion f:Γ→Δ and a compatible holomorphic embedding F:H→Hr that is with F(yZ)=f(y)F(z) for all γ∈Γ and all z∈H

We classify all one-class genera of admissible lattice chains of length at least 2 in hermitian spaces over number fields. If L is a lattice in the chain and \(\mathfrak {p} \) the prime ideal dividing the index of the lattices in the chain, then the \(\{ \mathfrak {p} \} \)-arithmetic group \(\mathrm {Aut} (L_{ \{ \mathfrak {p} \} }) \) acts chamber transitively on the corresponding Bruhat-Tits building. So our classification provides a step forward to a complete classification of these chamber transitive groups which has been announced 1987 (without a detailed proof) by Kantor, Liebler and Tits. In fact we find all their groups over number fields and one additional building with a discrete chamber transitive group.

Length spectra of periodic orbits are investigated for some chaotic dynamical systems whose quantum energy spectra show unexpected statistical properties and for which the notion of arithmetical chaos has been introduced recently. These systems are defined as the unconstrained motions of particles on two-dimensional surfaces of constant negative curvature whose fundamental groups are given by number theoretical statements (arithmetic Fuchsian groups). It is shown that the mean multiplicity of lengths l of periodic orbits grows asymptotically like c.e12//l,I to infinity ., Moreover, the constant c (depending on the arithmetic group) is determined.

Let X X be a smooth curve over a finite field of characteristic p p , let E E be a number field, and let L = { L λ } \mathbf {L} = \{\mathcal {L}_\lambda \} be an E E -compatible system of lisse sheaves on the curve X X . For each place λ \lambda of E E not lying over p p , the λ \lambda -component of the system L \mathbf {L} is a lisse E λ E_\lambda -sheaf L λ \mathcal {L}_\lambda on X X , whose associated arithmetic monodromy group is an algebraic group over the local field E λ E_\lambda . We use Serre’s theory of Frobenius tori and Lafforgue’s proof of Deligne’s conjecture to show that when the E E -compatible system L \mathbf {L} is semisimple and pure of some integer weight, the isomorphism type of the identity component of these monodromy groups is “independent of λ \lambda ”. More precisely, after replacing E E by a finite extension, there exists a connected split reductive algebraic group G 0 G_0 over the number field E E such that for every place λ \lambda of E E not lying over p p , the identity component of the arithmetic monodromy group of L λ \mathcal {L}_\lambda is isomorphic to the group G 0 G_0 with coefficients extended to the local field E λ E_\lambda .

We investigate sections of the arithmetic fundamental groups of hyperbolic curves over function fields. As a consequence we prove that the anabelian section conjecture of Grothendieck holds over all finitely generated fields over \mathbb Q if it holds over all number fields, under the condition of finiteness (of the \mathcal l -primary parts) of certain Shafarevich–Tate groups. We also prove that if the section conjecture holds over all number fields then it holds over all finitely generated fields for curves which are defined over a number field.

Let G be a reductive algebraic group over Q and Γ⊂G(Q) an arithmetic subgroup. Let K∞⊂G(R) be a maximal compact subgroup. We study the asymptotic behavior of the counting functions of the cuspidal and residual spectrum, respectively, of the regular representation of G(R) in L2(Γ﹨G(R)) of a fixed K∞-type σ. A conjecture, which is due to Sarnak, states that the counting function of the cuspidal spectrum of type σ satisfies Weyl's law and the residual spectrum is of lower order growth. Using the Arthur trace formula we reduce the conjecture to a problem about L-functions occurring in the constant terms of Eisenstein series. If G satisfies property (L), introduced by Finis and Lapid, we establish the conjecture. This includes classical groups over a number field.

We prove that the number of unimodular integral n × n matrices in a norm ball whose characteristic polynomial has Galois group different than the full symmetric group Sn is of strictly lower order of magnitude than the number of all such matrices in the ball, as the radius increases. More generally, we prove a similar result for the Galois groups associated with elements in any connected semisimple linear algebraic group defined and simple over a number field F . Our method is based on the abstract large sieve method developed by Kowalski, and the study of Galois groups via reductions modulo primes developed by Jouve, Kowalski and Zywina. The two key ingredients are a uniform quantitative lattice point counting result, and a non-concentration phenomenon for lattice points in algebraic subvarieties of the group variety, both established previously by the authors. The results answer a question posed by Rivin and by Jouve, Kowalski and Zywina, who have considered Galois groups of random products of elements in algebraic groups.

We show that the Halphen transform of a Lamé equation can be written as the symmetric square of the Lamé equation followed by an Euler transform. We use this to compute a list of Lamé equations with arithmetic Fuchsian monodromy group. It contains all those Lamé equations where the quaternion algebra A associated to the arithmetic Fuchsian group is defined over ℚ.

In this paper we prove local–global principles for the existence of an embedding (E, σ) ↪ (A, τ) of a given global field E endowed with an involutive automorphism σ into a simple algebra A given with an involution τ in all situations except where A is a matrix algebra of even degree over a quaternion division algebra and τ is orthogonal (Theorem A of the introduction). Rather surprisingly, in the latter case we have a result which in some sense is opposite to the local–global principle, viz. algebras with involution locally isomorphic to (A, τ) are distinguished by their maximal subfields invariant under the involution (Theorem B of the introduction). These results can be used in the study of classical groups over global fields. In particular, we use Theorem B to complete the analysis of weakly commensurable Zariski-dense S -arithmetic groups in all absolutely simple algebraic groups of type different from D_4 which was initiated in our paper [23]. More precisely, we prove that in a group of type D_n , n even > 4 , two weakly commensurable Zariski-dense S -arithmetic subgroups are actually commensurable. As indicated in [23], this fact leads to results about length-commensurable and isospectral compact arithmetic hyperbolic manifolds of dimension 4n + 7 , with n ≥ 1 . The appendix contains a Galois-cohomological interpretation of our embedding theorems.

We establish some geometric constraints on compact Coxeter polytopes in hyperbolic spaces and show that these constraints can be a very useful tool for the classification problem of reflective anisotropic Lorentzian lattices and cocompact arithmetic hyperbolic reflection groups.

Suppose is a simple and simply connected algebraic group over an algebraic number field and is a finite set of valuations of containing all Archimedean valuations. This paper is a study of the connections between abstract properties of the -arithmetic subgroup and the congruence property, i.e. the finiteness of the corresponding congruence kernel . In particular, it is shown that if the profinite completion satisfies condition (PG), (i.e., for any integer > 0 and any prime there exist and such that for all ), then is finite. Examples are given demonstrating the possibility of effectively verifying (PG)' .

Given an $S$-arithmetic group, we ask how much information on the ambient algebraic group, number field of definition, and set of places $S$ is encoded in the commensurability class of the profinite completion. As a first step, we show that the profinite

Given an abelian variety over a number field, its Sato–Tate group is a compact Lie group which conjecturally controls the distribution of Euler factors of the L L -function of the abelian variety. It was previously shown by Fité, Kedlaya, Rotger, and Sutherland that there are 52 groups (up to conjugation) that occur as Sato–Tate groups of abelian surfaces over number fields; we show here that for abelian threefolds, there are 410 possible Sato–Tate groups, of which 33 are maximal with respect to inclusions of finite index. We enumerate candidate groups using the Hodge-theoretic construction of Sato–Tate groups, the classification of degree-3 finite linear groups by Blichfeldt, Dickson, and Miller, and a careful analysis of Shimura’s theory of CM types that rules out 23 candidate groups; we cross-check this using extensive computations in GAP, SageMath, and Magma. To show that these 410 groups all occur, we exhibit explicit examples of abelian threefolds realizing each of the 33 maximal groups; we also compute moments of the corresponding distributions and numerically confirm that they are consistent with the statistics of the associated L L -functions.

An algebraic description of basic discrete symmetries (space reversal P, time reversal T, and their combination PT) is studied. Discrete subgroups of orthogonal groups of multidimensional spaces over the fields of real and complex numbers are considered in terms of fundamental automorphisms of Clifford algebras. In accordance with a division ring structure, a complete classification of automorphism groups is established for the Clifford algebras over the field of real numbers. The correspondence between eight double coverings (Dąbrowski groups) of the orthogonal group and eight types of the real Clifford algebras is defined with the use of isomorphisms between the automorphism groups and finite groups. Over the field of complex numbers there is a correspondence between two nonisomorphic double coverings of the complex orthogonal group and two types of complex Clifford algebras. It is shown that these correspondences associate with a well-known Atiyah–Bott–Shapiro periodicity. Generalized Brauer–Wall groups are introduced on the extended sets of the Clifford algebras. The structure of the inequality between the two Clifford–Lipschitz groups with mutually opposite signatures is elucidated. The physically important case of the two different double coverings of the Lorentz groups is considered in details.