Arithmetic Groups Research Articles

Analytic torsion for arithmetic locally symmetric manifolds and approximation of L2-torsion

In this paper we define a regularized version of the analytic torsion for quotients of a symmetric space of non-positive curvature by arithmetic lattices. The definition is based on the study of the renormalized trace of the corresponding heat operators, which is defined as the geometric side of the Arthur trace formula applied to the heat kernel. Then we study the limiting behavior of the analytic torsion as the lattices run through a sequence of congruence subgroups of a fixed arithmetic subgroup. Our main result states that for sequences of principal congruence subgroups, which converge to 1 at a fixed finite set of places and strongly acyclic flat bundles, the logarithm of the analytic torsion, divided by the index of the subgroup, converges to the L2-analytic torsion.

Residual finiteness of extensions of arithmetic subgroups of {{,mathrm{SU},}}(d,1) with cusps

Let Gamma be an arithmetic subgroup of {{,mathrm{SU},}}(d,1) with cusps, and let X_Gamma be the associated locally symmetric space. In this paper we investigate the pre-image of Gamma in the covering groups of {{,mathrm{SU},}}(d,1). Let H^bullet _!(X_Gamma ,mathbb {C}) be the inner cohomology, i.e. the image in H^bullet (X_Gamma ,mathbb {C}) of the compactly supported cohomology. We prove that if the first inner cohomology group H^1_!(X_Gamma ,mathbb {C}) is non-zero then the pre-image of Gamma in each connected cover of {{,mathrm{SU},}}(d,1) is residually finite. At the end of the paper we give an example of an arithmetic subgroup Gamma satisfying the criterion H^1_!(X_Gamma ,mathbb {C}) ne 0.

Rotation equivalence and cocycle superrigidity

We analyze Euclidean spheres in higher dimensions and the corresponding orbit equivalence relations induced by the group of rational rotations from the viewpoint of descriptive set theory. It turns out that such equivalence relations are not treeable in dimension greater than $2$. Then we show that the rotation equivalence relation in dimension $n \geq 5$ is not Borel reducible to the one in any lower dimension. Our methods combine a cocycle superrigidity result from the works of Furman and Ioana with the superrigidity theorem for $S$-arithmetic groups of Margulis. We also apply our techniques to give a geometric proof of the existence of uncountably many pairwise incomparable equivalence relations up to Borel reducibility.

PEMETAAN DAN PENGEMBANGAN POTENSI DESA MATA WOGA DALAM BIDANG ADMINISTRASI, PENDIDIKAN, PERIKANAN, DAN PERTANIAN


 
 
 
 Kuliah Kerja Nyata (KKN) is a form of community service program held by universities in Indonesia. In KKN Program held in Desa Mata Woga, Kecamatan Katiku Tana, Kabupaten Sumba Tengah, the students and lecture of Unkriswina Sumba carried out various activities with the aim of mapping and developing village potential. The purpose of this KKN activity is to answer various social issues found, among others, in the management of village administration, education, health, and agriculture. There are 4 methods of carried out, (1) population data collection and village website creation, (2) improvement of basic literacy in the form of reading and arithmetic learning groups, (3) freshwater fish cultivation, and (4) production of bokasi fertilizer. The result of this activity is the availability of accurate and complete population data and the availability of a village website that can be used to present information about Mata Woga. In addition, in terms of developing community potential, the results obtained are an increase in children's reading and numeracy skills, the availability of freshwater fish ponds which are also expected to help improve community nutrition and prevent and accelerate the decline in stunting rates, and increase public knowledge in the use of waste by making bokasi fertilizer. In addition, the KKN group is also involved in various other activities, such making trash cans, nameplate making, building vegetable beds, posyandu and PKK activities, and cleaning the village environment. The results of the activities above show that this KKN program has been able to answer some of the problems that exist in the village of Mata Woga related to the mapping of village potentials and the development of existing potentials in the fields of education, health and fishery, as well as agriculture.
 
 
 

Arbeitsgemeinschaft: Derived Galois Deformation Rings and Cohomology of Arithmetic Groups

The purpose of the workshop was to study derived generalizations of Mazur’s deformation ring of Galois representations, and the relationship of such a derived deformation ring to the homology of arithmetic groups.

Improved computation of fundamental domains for arithmetic Fuchsian groups

A practical algorithm to compute the fundamental domain of an arithmetic Fuchsian group was given by Voight, and implemented in Magma. It was later expanded by Page to the case of arithmetic Kleinian groups. We combine and improve on parts of both algorithms to produce a more efficient algorithm for arithmetic Fuchsian groups. This algorithm is implemented in PARI/GP, and we demonstrate the improvements by comparing running times versus the live Magma implementation.

Quotients of the Bruhat-Tits tree by arithmetic subgroups of special unitary groups

Let $K$ be the function field of a curve $C$ over a field $\mathbb{F}$ of either odd or zero characteristic. Following the work by Serre and Mason on $\mathrm{SL}_2$, we study the action of arithmetic subgroups of $\mathrm{SU}(3)$ on its corresponding Bruhat-Tits tree associated to a suitable completion of $K$. More precisely, we prove that the quotient graph "looks like a spider", in the sense that it is the union of a set of cuspidal rays (the "legs"), parametrized by an explicit Picard group, that are attached to a connected graph (the "body"). We use this description in order to describe these arithmetic subgroups as amalgamated products and study their homology. In the case where $\mathbb{F}$ is a finite field, we use a result by Bux, K\"ohl and Witzel in order to prove that the "body" is a finite graph, which allows us to get even more precise applications.

The Semi-absolute Anabelian Geometry of Geometrically Pro-$p$ Arithmetic Fundamental Groups of Associated Low-Dimensional Configuration Spaces

Let p be a prime number. In the present paper, we study geometrically pro- p arithmetic fundamental groups of low-dimensional configuration spaces associated to a given hyperbolic curve over an arithmetic field such as a number field or a p -adic local field. Our main results concern the group-theoretic reconstruction of the function field of certain tripods (i.e., copies of the projective line minus three points) that lie inside such a configuration space from the associated geometrically pro- p arithmetic fundamental group, equipped with the auxiliary data constituted by the collection of decomposition groups determined by the closed points of the associated compactified configuration space.

Application of automorphic forms to lattice problems

Abstract In this article, we propose a new approach to the study of lattice problems used in cryptography. We specifically focus on module lattices of a fixed rank over some number field. An essential question is the hardness of certain computational problems on such module lattices, as the additional structure may allow exploitation. The fundamental insight is the fact that the collection of those lattices are quotients of algebraic manifolds by arithmetic subgroups. Functions in these spaces are studied in mathematics as part of the number theory. In particular, those form a module over the Hecke algebra associated with the general linear group. We use results on these function spaces to define a class of distributions on the space of lattices. Using the Hecke algebra, we define Hecke operators associated with collections of prime ideals of the number field and show a criterion on distributions to converge to the uniform distribution, if the Hecke operators are applied to the chosen distribution. Our approach is motivated by the work of de Boer, Ducas, Pellet-Mary, and Wesolowski (CRYPTO’20) on self-reduction of ideal lattices via Arakelov divisors.

Extension of period maps by polyhedral fans

Kato and Usui developed a theory of partial compactifications for quotients of period domains D by arithmetic groups Γ, in an attempt to generalize the toroidal compactifications of Ash-Mumford-Rapoport-Tai to non-classical cases. Their partial compactifications, which aim to fully compactify the images of period maps, rely on the choice of a fan which is strongly compatible with Γ. In particular, they conjectured the existence of a complete fan, which would serve to simultaneously compactify all period maps of a given type.In this article, we briefly review the theory, and construct a fan which compactifies the image of a period map arising from a particular two-parameter family of Calabi-Yau threefolds studied by Hosono and Takagi, with Hodge numbers (1,2,2,1). On the other hand, we disprove the existence of complete fans in some general cases, including the (1,2,2,1) case.

Can training in the approximate number system improve the informal mathematics ability of preschoolers?

Recent studies show that comparison or arithmetic training in the approximate number system (ANS) can improve the early mathematics ability of preschool children. However, no studies have compared the training effects of ANS comparison training with those of ANS arithmetic training on the early mathematics ability of preschool children. The current study pseudorandomly assigned 87 children aged 4–5 years to one of three training groups (the ANS comparison, ANS arithmetic, and control groups) for 4 weeks of training. The results showed that compared with the control group, the ANS comparison training and ANS arithmetic training equally improved the ANS acuity and informal mathematics ability of preschool children. In addition, the study found that there may be a bidirectional causal relationship between ANS and mathematics in preschoolers, but this relationship needs to be further investigated using longitudinal studies. Taken together, these findings emphasize the importance of ANS-based training in improving preschoolers' ANS acuity and informal mathematics ability before formal school enrollment.

FROM GEOMETRY TO ARITHMETIC OF COMPACT HYPERBOLIC COXETER POLYTOPES

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.

Perturbative higher-derivative terms in mathcal{N} = 6 asymmetric orbifolds

We analyse the perturbative four-point amplitudes in the simplest string theory examples of T-fold backgrounds, which enjoy mathcal{N} = 6 supersymmetries in four dimensions. There are two theories defined as asymmetric orbifolds of order 2 and 3, respectively. The perturbative spectrum and the one-loop four-point amplitudes are shown to be invariant under an arithmetic symplectic group defined over the Hurwitz (respectively Eisenstein) quaternions. The supersymmetry constraints on the low energy effective couplings are determined and we conjecture the U-duality group of the theory and the non-perturbative completion of the six-derivative coupling as a rank one theta series. We discuss the non-perturbative spectrum of BPS states in the light of our analysis.

Charmenability of arithmetic groups of product type

We discuss special properties of the spaces of characters and positive definite functions, as well as their associated dynamics, for arithmetic groups of product type. Axiomatizing these properties, we define the notions of charmenability and charfiniteness and study their applications to the topological dynamics, ergodic theory and unitary representation theory of the given groups. To do that, we study singularity properties of equivariant normal ucp maps between certain von Neumann algebras. We apply our discussion also to groups acting on product of trees.

A note on non-ordinary primes for some genus-zero arithmetic groups

Suppose that OL is the ring of integers of a number field L, and suppose thatf(z)=∑n=1∞af(n)qn∈Sk(Γ0(N)+)∩OL[[q]] is a normalized Hecke eigenform for Γ0(N)+. We say that f is non-ordinary at p if there is a prime ideal p⊂OL above p for which af(p)≡0(modp). In the authors' previous paper with Ken Ono [10] it was proved that there are infinitely many Hecke eigenforms for SL2(Z) such that are non-ordinary at any given finite set of primes. In this paper, we extend this result to some genus 0 subgroups of SL2(R), namely, the normalizers Γ0(N)+ of the congruence subgroups Γ0(N). Our result also generalizes some of Choi and Kim's result in [2].

A dichotomy for integral group rings via higher modular groups as amalgamated products

We show that U(ZG), the unit group of the integral group ring ZG, either satisfies Kazhdan's property (T) or is, up to commensurability, a non-trivial amalgamated product, in case G is a finite group satisfying some mild conditions. A key step in the proof is the construction of amalgamated decompositions of the elementary group E2(O), where O is an order in rational division algebra, and of certain arithmetic groups Γ. The methods for the latter turn out to work in much greater generality and most notably are carried out to obtain amalgam decompositions for the higher modular groups SL+(Γn(Z)), with n≤4, which can be seen as higher dimensional versions of modular and Bianchi groups. For this we introduce a subgroup mimicking the elementary linear group, denoted E2(Γn(Z)). We prove that E2(Γn(Z)) has always a non-trivial decomposition as a free product with amalgamated subgroup E2(Γn−1(Z)).

Asymptotics for Hecke eigenvalues of automorphic forms on compact arithmetic quotients

In this paper, we describe the asymptotic distribution of Hecke eigenvalues in the Laplace eigenvalue aspect for certain families of Hecke-Maass forms on compact arithmetic quotients. Instead of relying on the trace formula, which was the primary tool in preceding studies on the subject, we use Fourier integral operator methods. This allows us to treat not only spherical, but also non-spherical Hecke-Maass forms with corresponding remainder estimates. Our asymptotic formulas are available for arbitrary simple and connected algebraic groups over number fields with cocompact arithmetic subgroups.

Boundary and Eisenstein cohomology of G_2(mathbb {Z})

In this article, Eisenstein cohomology of the arithmetic group G_2(mathbb {Z}) with coefficients in any finite dimensional highest weight irreducible representation has been determined. This is accomplished by studying the cohomology of the boundary of the Borel–Serre compactification.

Chudnovsky-Ramanujan type formulae for non-compact arithmetic triangle groups

We develop a uniform method to derive Chudnovsky-Ramanujan type formulae for triangle groups based on a generalization of a method of Chudnovsky and Chudnovsky; in particular, we carry out the method systematically for non-compact arithmetic triangle groups and one non-Fuchsian covering. As a result, we derive all rational Ramanujan type series given by Chan-Cooper for levels 1-4, as well as two additional rational series of a similar form prescribed by Chan-Cooper for these levels, but not found in the paper of Chan-Cooper. These two additional series were first found by Z.-W. Sun in a slightly different form. We also derive additional rational series of a similar form, but not found in the papers of Chan-Cooper nor Z.-W. Sun.As an ingredient in the method, we give an algorithm to rigorously confirm the singular values of normalized Eisenstein series of weight 2 and give examples in higher class number, which may be of independent interest. We also prove some partial completeness results for Clausen type single summation series.

On quaternionic rigid meromorphic cocyles

Recently, Darmon and Vonk initialized the theory of rigid meromorphic cocycles for the group $\mathrm{SL}_2(\mathbb{Z}[1/p])$. One of their major results is the algebraicity of the divisor associated to such a cocycle. We generalize the result to the setting of $\mathfrak{p}$-arithmetic subgroups of inner forms of $\mathrm{SL}_2$ over arbitrary number fields. The method of proof differs from the one of Darmon and Vonk. Their proof relies on an explicit description of the cohomology via modular symbols and continued fractions, whereas our main tool is Bieri-Eckmann duality for arithmetic groups.