Height Estimates for Bianchi Groups

Here we include some classical results from the theory of binary hermitian forms which originate from Hermite (1854). We discuss the reduction theory of binary hermitian forms as described for example in Bianchi (1892). Eventually our considerations lead to Humbert’s computation of the covolume of SL(2, Thuong) where Thuong is the ring of integers in an imaginary quadratic number field. The work of Humbert on hermitian forms is contained in his papers (1915), (1919a)—(1919e). It contains an interesting error, we correct it in Section 9.6. We also develop a theory of representation numbers of binary hermitian forms which is analogous to the theory of binary quadratic forms as in Landau (1927).KeywordsPrime NumberPrime IdealZeta FunctionDirichlet SeriesHermitian FormThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

On a question of Serre

Denote by Q(sqrt{-m}), with m a square-free positive integer, an imaginary quadratic number field, and by A its ring of integers. The Bianchi groups are the groups SL_2(A). We reveal a correspondence between the homological torsion of the Bianchi groups and new geometric invariants, which are effectively computable thanks to their action on hyperbolic space. We expose a novel technique, the torsion subcomplex reduction, to obtain these invariants. We use it to explicitly compute the integral group homology of the Bianchi groups. Furthermore, this correspondence facilitates the computation of the equivariant K-homology of the Bianchi groups. By the Baum/Connes conjecture, which is verified by the Bianchi groups, we obtain the K-theory of their reduced C*-algebras in terms of isomorphic images of their equivariant K-homology.

We study a family of functions defined in a very simple way as sums of powers of binary Hermitian forms with coefficients in the ring of integers of an Euclidean imaginary quadratic field K with discriminant dK. Using these functions we construct a nontrivial cocycle belonging to the space of parabolic cocycles on Euclidean Bianchi groups. We also show that the average value of these functions is related to the special values of L(χdK,s). Using the properties of these functions we give new and computationally efficient formulas for computing some special values of L(χdK,s).

For imaginary quadratic number fields F = <TEX>$\mathbb{Q}(\sqrt{{\varepsilon}p_1{\ldots}p_{t-1}})$</TEX>, where <TEX>${\varepsilon}{\in}$</TEX>{-1,-2} and distinct primes <TEX>$p_i{\equiv}1$</TEX> mod 4, we give condition of 8-ranks of class groups C(F) of F equal to 1 or 2 provided that 4-ranks of C(F) are at most equal to 2. Especially for F = <TEX>$\mathbb{Q}(\sqrt{{\varepsilon}p_1p_2)$</TEX>, we compute densities of 8-ranks of C(F) equal to 1 or 2 in all such imaginary quadratic fields F. The results are stated in terms of congruence relation of <TEX>$p_i$</TEX> modulo <TEX>$2^n$</TEX>, the quartic residue symbol <TEX>$(\frac{p_1}{p_2})4$</TEX> and binary quadratic forms such as <TEX>$p_2^{h+(2_{p_1})/4}=x^2-2p_1y^2$</TEX>, where <TEX>$h+(2p_1)$</TEX> is the narrow class number of <TEX>$\mathbb{Q}(\sqrt{2p_1})$</TEX>. The results are also very useful for numerical computations.

Abstract

Some useful information is known about the fundamental domain for certain Hilbert modular groups. The six nonequivalent points with nontrivial isotropy in the fundamental domains under the action of the modular group for Q(5), Q(2), and Q(3) have been determined previously by Gundlach. In finding these points, use was made of the exact size of the isotropy groups. Here we show that the fixed points and the isotropy groups can be found without such knowledge by use of a computer scan. We consider the cases Q(5) and Q(2). A computer algebra system and a C compiler were essential in perfoming the computations.

We strongly develop the relationship between complex hyperbolic geometry and arithmetic counting or equidistribution applications, that arises from the action of arithmetic groups on complex hyperbolic spaces, especially in dimension $2$. We prove a Mertens' formula for the integer points over a quadratic imaginary number fields $K$ in the light cone of Hermitian forms, as well as an equidistribution theorem of the set of rational points over $K$ in Heisenberg groups. We give a counting formula for the cubic points over $K$ in the complex projective plane whose Galois conjugates are orthogonal and isotropic for a given Hermitian form over $K$, and a counting and equidistribution result for arithmetic chains in the Heisenberg group when their Cygan diameter tends to $0$.

We develop methods for the rapid computation of the regulator of a real quadratic congruence function field K = k(x)(√D). By extending Shanks' infrastructure ideas in real quadratic number fields to real quadratic congruence function fields we obtain a baby step–giant step method for evaluating the regulator of K in O(|D|¼) polynomial operations. We also show the existence of an effective algorithm which computes the regulator unconditionally in O(|D|⅕) polynomial operations. By implementing both methods on a computer, we found that the O(|D|⅕) algorithm tends to be far better than the baby step–giant step algorithm in those cases where the regulator exceeds 108.

This is a report on our work during the last few years on extending the Bieri-Neumann-Strebel-Renz theory of “geometric invariants” of groups to a theory of group actions on non-positively curved (= CAT(0)) spaces. With the exception of Theorem 8, which is proved here, and the related material in §5.3, proofs of all our theorems can be found in our papers [BG I ] (controlled connectivity and openness results), [BG II ] (the geometric invariants) and [BG III ] ( SL 2 actions on the hyperbolic plane). An earlier expository paper [BG 98] is also relevant. The geometric invariants we recall the “geometric” or “Σ-” invariants of groups developed during the 1980's by Bieri, Neumann, Strebel and Renz (abbrev. BNSR); see [BNS 87], [BR 88], [Re 88]. We set things out in a way which leads directly to generalizations which were not anticipated in the original literature. Let G be a group of type F n , n ≥ 1. Let X be a contractible G -CW complex which is either (a) free with cocompact n -skeleton, or (b) properly discontinuous and cocompact. Case (a) exists by the definition of F n ; Case (b) is often useful but can only exist when G has finite virtual cohomological dimension. Controlled connectivity Let χ : G → ℝ be a non-zero character, i.e., a homomorphism to the additive group of real numbers. Reinterpret ℝ as the group of translations, Transl, of the Euclidean line, and thus reinterpret χ as an action of G on by translations.

A method is developed for finding all G-modules (where G is a connected and simply connected semisimple algebraic group over an algebraically closed field of characteristic zero) whose algebra of invariants has prescribed homological dimension. The main theorem says that the number of such G-modules, considered to within isomorphism and addition of a trivial direct summand, is finite. The same result is proved for finite groups G. All algebras of invariants of homological dimension ?10 of a single binary form are found, as well as all algebras of invariants of a system of binary forms that are hypersurfaces. It is shown that the exceptional simple groups have no irreducible modules with an algebra of invariants of small nonzero homological dimension. Bibliography: 46 titles.

Let G denote the finite cyclic group of order n. The problem of determining necessary and sufficient conditions for F to be the fixed point set of a smooth cyclic group action on some sphere has been solved when n is a prime power. P. A. Smith proved that F must be a Zn-homology sphere. If n is odd it is also known that F is unitary. L. Jones has shown that these conditions are also sufficient to realize F as the fixed point set of smooth cyclic group action on some sphere when n is a prime power Pl. In the general case it is known that F is a union of smooth manifolds, unitary if n is odd. And if Zn acts on some even dimensional sphere, then the Euler characteristic number of F is 2 if action is orientation-preserving and 0 if action is orientation-reversing. We may ask about possible restrictions on Pontryagin numbers of components of the fixed set F. If n is a prime power, the Pontryagin numbers of the fixed point set all vanish by the P. A. Smith theorem and the Hirzebruch signature theorem. It is natural to ask whether such restrictions hold for other types of smooth cyclic groups acting on spheres. In case where n is not a prime power, by work of R. Oliver [01] and its extention of A. Assadi [Asd] and K. Pawalowski [Pa 1-4], we can construct exotic actions on spheres such that Pontryagin classes of fixed point sets do not vanish. But in these examples the Pontryagin numbers all vanish because these actions bound group actions on disks. R. Schultz has shown in [S2] that if G is a cyclic group whose order is not a prime power, then there are smooth actions of G on spheres such that the fixed point sets have nonzero Pontryagin numbers provided the dimension of the fixed point set is greater than 16. Testing to see if the lower bound in dimensions is necessary. There are two possibilities. First, more sophisticated computations might make it possible to remove the restriction on dimensions. Second, there might be some unusual things happening in low dimensions (compare Ewing's result for Zp actions on spheres). Main Theorem is evidence for the first one.

In recently published work Maskit constructs a fundamental domain D_g for the Teichmueller modular group of a closed surface S of genus g>1. Maskit's technique is to demand that a certain set of 2g non-dividing geodesics C_{2g} on S satisfies certain shortness criteria. This gives an a priori infinite set of length inequalities that the geodesics in C_{2g} must satisfy. Maskit shows that this set of inequalities is finite and that for genus g=2 there are at most 45. In this paper we improve this number to 27. Each of these inequalities: compares distances between Weierstrass points in the fundamental domain S-C_4 for S; and is realised (as an equality) on one or other of two special surfaces.

We will complete the list of universal binary Hermitian forms over imaginary quadratic fields by investigating three Hermitian forms missed by previous researchers.

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