Showing papers on "Arithmetic group published in 2017"
TL;DR: In this paper it was shown that up to scaling there are only finitely many integral lattices L of signature (2,n) with n>20 or n=17 such that the modular variety defined by the orthogonal group of L is not of general type.
Abstract: We prove that up to scaling there are only finitely many integral lattices L of signature (2,n) with n>20 or n=17 such that the modular variety defined by the orthogonal group of L is not of general type. In particular, when n>107, every modular variety defined by an arithmetic group for a rational quadratic form of signature (2,n) is of general type. We also obtain similar finiteness in n>8 for the stable orthogonal groups. As a byproduct we derive finiteness of lattices admitting reflective modular form of bounded vanishing order, which proves a conjecture of Gritsenko and Nikulin.
14 citations
TL;DR: In this article, the authors gave a list of 51 monodromy groups corresponding to primitive pairs of degree four polynomials, which are products of cyclotomic polynomial, and for which, the absolute value of the leading coefficient of the difference poynomial is greater than 2.
11 citations
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TL;DR: In this article, a cohomological interpretation of the Eichler-Shimura isomorphism as a connection morphism of certain exact sequence of G-modules, for the multiplicative group G of any quaternion algebra over a totally real field, is given.
Abstract: The Eichler-Shimura isomorphism realizes the automorphic representation generated by an automorphic newform in certain cohomology of an arithmetic group. In this short note, we give a cohomological interpretation of the Eichler-Shimura isomorphism as a connection morphism of certain exact sequence of G-modules, for the multiplicative group G of any quaternion algebra over a totally real field.
5 citations
TL;DR: In this article, the level of the maximal principal congruence subgroup in an arithmetic group is computed using a GAP implementation of the algorithms, which is the main component of our methods for computing with such arithmetic groups.
Abstract: For $n > 2$, let $\Gamma$ denote either $SL(n, Z)$ or $Sp(n, Z)$. We give a practical algorithm to compute the level of the maximal principal congruence subgroup in an arithmetic group $H\leq \Gamma$. This forms the main component of our methods for computing with such arithmetic groups $H$. More generally, we provide algorithms for computing with Zariski dense groups in $\Gamma$. We use our GAP implementation of the algorithms to solve problems that have emerged recently for important classes of linear groups.
TL;DR: In this article, it was shown that the Chern character commutes with Hecke operators and that the Shimura product of double cosets naturally corresponds to the Kasparov product.
Abstract: Let $\Gamma$ be a lattice in a locally compact group $G$. In earlier work, we used $KK$-theory to equip the $K$-groups of any $\Gamma$-$C^{*}$-algebra on which the commensurator of $\Gamma$ acts with Hecke operators. When $\Gamma$ is arithmetic, this gives Hecke operators on the $K$-theory of certain $C^{*}$-algebras that are naturally associated with $\Gamma$. In this paper, we first study the topological $K$-theory of the arithmetic manifold associated to $\Gamma$. We prove that the Chern character commutes with Hecke operators. Afterwards, we show that the Shimura product of double cosets naturally corresponds to the Kasparov product and thus that the $KK$-groups associated to an arithmetic group $\Gamma$ become true Hecke modules. We conclude by discussing Hecke equivariant maps in $KK$-theory in great generality and apply this to the Borel-Serre compactification as well as various noncommutative compactifications associated with $\Gamma$. Along the way we discuss the relation between the $K$-theory and the integral cohomology of low-dimensional manifolds as Hecke modules.
TL;DR: In this article, it was shown that the SATake-Baily-Borel compactification of an arithmetic quotient of a symmetric domain of type IV is smooth at infinity if and only if the group is generated by reflections and the fundamental polyhedral cone of the group dΓ in the future cone is a simplicial cone.
Abstract: Let Γ be an arithmetic group of affine automorphisms of the n-dimensional future tube T. It is proved that the quotient space T/Γ is smooth at infinity if and only if the group Γ is generated by reflections and the fundamental polyhedral cone (“Weyl chamber”) of the group dΓ in the future cone is a simplicial cone (which is possible only for n ≤ 10). As a consequence of this result, a smoothness criterion for the Satake–Baily–Borel compactification of an arithmetic quotient of a symmetric domain of type IV is obtained.