Jens Mennicke

On the groups SL2(ℤ[x]) and SL2(k[x, y])

This paper studies free quotients of the groups SL2(ℤ[x]) and SL2(k[x, y]),k a finite field. These quotients give information about the relation of the above groups to their subgroups generated by elementary or unipotent elements.

Zeta-functions of binary Hermitian forms and special values of Eisenstein series on three-dimensional hyperbolic space

The present work contains a systematic investigation of zeta-functions of binary Hermitian forms over the ring o of integers of an imaginary quadratic number field K = tl~(]/~) with discriminant D < 0. The resulting relations between these zetafunctions have numerous applications to the computation of special values of the Eisenstein series for the group PSL(2,o) acting on the three-dimensional hyperbolic space I-I. In particular we reprove many known formulae for representation numbers for quaternary quadratic forms over Z, and we obtain many explicit new results for such representation numbers. In Sect. 2 we develop a theory of representations of numbers by binary Hermitian forms over o which parallels the classical theory of representations of integers by binary quadratic forms over ~. A key result is a certain bijection described in Theorem 2.3. This bijection means that the sum of the numbers of representations [modulo SL(2, o)-units] of an integer k # 0 by a set of representatives of the SL(2, o)-classes of binary Hermitian forms over o with discriminant A is equal to the number of cosets 2 +ko with 2~ o, 22-+ A = 0 m o d k . The number of these cosets is explicitly computed in Sect. 2 in all cases. Hence a certain mean value of the representation numbers is known whereas simple formulae for the individual representation numbers do not exist. We apply the formulae of Sect. 2 to the study of zeta-functions of (definite or indefinite) Hermitian forms. For simplicity we restrict in this introduction to the case of positive definite forms over o although the results of Chap. I are valid in the indefinite case as well. Let f be a positive definite binary Hermitian form over o with SL(2, o)-unit group gl(f) . Then we define

Groups Acting Discontinuously on Three-Dimensional Hyperbolic Space

In this part we shall describe fundamental facts from the theory of transformation groups on 3-dimensional hyperbolic space. The more elementary theory is quoted from Ford (1951) or Beardon (1977), (1983) We set up the theory as far as is necessary for the chapters to come. We can only quote the more difficult theorems on the subject. To include the proofs of all of them would have blown up the size of this book.

Three-Dimensional Hyperbolic Space

Three-dimensional hyperbolic space is the unique 3-dimensional connected and simply connected Riemannian manifold with constant sectional curvature equal to —1. This space has certain concrete models which all have certain advantages. We discuss here the four most classical ones.

Spectral Theory of the Laplace Operator for Cofinite Groups

This chapter is a continuation of Chapter 4. Having established there some fundamental facts about the spectral theory of the Laplace operator on the Hilbert spaces L2 (Γ\IH) for discrete subgroups Γ < PSL(2, C) and having treated the case of cocompact groups in Chapter 5, we turn here to the finer properties of the Laplace operator for groups Γ which are of finite covolume but not cocompact. We already know that Δ defined on an appropriate domain in L2 (Γ\IH) has a unique self-adjoint extension Δ. If Γ is not cocompact the spectrum of Δ is no longer purely discrete. We aim here for an explicit description of the non-discrete part of the spectrum of Δ and also for an explicit decomposition formula for L2-functions. We shall give two versions of this theory in Section 3. The first, more in the spirit of Selberg (1989b), gives Δ as a certain multiplication operator. The second describes the eigenpackets for Δ following Roelcke (1956a), (1966), (1967). Section 2 contains a quick treatment of the theory of eigenpackets. It is included here since nowadays textbooks treat the spectral theory of unbounded operators usually via spectral families.

Examples of Discontinuous Groups

This chapter contains various constructions for discontinuous groups of isometries of hyperbolic 3-space. Since we often use the terminology of Coxeter groups we report on this here. For the general theory of these groups see Bourbaki (1968).

Eisenstein Series for PSL(2) over Imaginary Quadratic Integers

Suppose that \(K = Q\left( {\sqrt D } \right) \) is an imaginary quadratic number field (D ∈ ℤ, D < 0 and square-free) and let Thuong be the ring of integers of K. We consider here the group PSL(2, Thuong) ⊂ PSL(2, ℂ). We already know from Chapter 7 that Γ is a discrete subgroup which is cofinite but not cocompact. We study the Eisenstein series defined in Chapter 3 in detail for the group PSL(2, Thuong). In fact we shall establish most of the general facts proved in Section 6.1 for the Eisenstein series of general cofinite groups by direct number theoretic methods. We shall for example relate the determinant of the scattering matrix to the zeta function of the Hilbert class field of K. The control we have over the Eisenstein series will also in turn imply many interesting number theoretic results.

Spectral Theory of the Laplace Operator

In the present chapter we analyse the operation of the Laplace-Beltrami operator on its natural domain in the Hilbert space L 2 (Г\IH) of squareintegrable functions on Г\IH for discrete subgroups Г < PSL(2, ℂ) in the frame-work of Hilbert space theory. Our discussion uses certain well-known facts about self-adjoint operators on Hilbert spaces, they can be found in Dunford, Schwartz (1958), Reed, Simon (1972) or Kato (1976).

Spectral Theory of the Laplace Operator for Cocompact Groups

In the present Chapter we give applications and finer points of the spectral theory of the Laplace-Beltrami operator Δ on L2(Γ\IH) in case Γ < PSL(2, ℂ) is a discrete cocompact group. We already know from the preceding Chapter that —Δ is essentially self-adjoint and positive on the subspace D ⊂ L2 (Γ\IH) consisting of all C 2-functions f ⋵ L 2(Γ\IH) such that Δf ∈ L 2 (Γ\IH) . This means that the closure of the graph of Δ in L 2 (Γ\IH) × L 2 (Γ\IH) is the graph of a self-adjoint linear operator \(\tilde \Delta :\tilde D \to {L^2}(\Gamma \backslash IH)\)

Integral Binary Hermitian Forms

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).