Die irreduziblen Darstellungen der Gruppen SL2(Zp), insbesondere SL2(Z2). I. Teil

Ohne Zusammenfassung Unterstützt durch den Schweizerischen Nationalfonds (820.167.73).     

Asymptotically invariant sequences and an action of SL (2,Z) on the 2-sphere

LetG be a countable group which acts non-singularly and ergodically on a Lebesgue space (X, ȑ, μ). A sequence (B _n) in ℒ is calledasymptotically invariant in lim _n μ (B _nΔgB _n)=0 for everygεG. In this paper we show that the existence of such sequences can be characterized by certain simple assumptions on the cohomology of the action ofG onX. As an explicit example we prove that a natural action of SL (2,Z) on the 2-sphere has no asymptotically invariant sequences. The last section deals with a particular cocycle for this action which has an interpretation as a random walk on the integers with “time” in SL (2,Z).     

Die irreduziblen Darstellungen der Gruppen SL2(Zp), insbesondere SL2(Z2). II. Teil

Ohne Zusammenfassung Unterstützt durch den Schweizerischen Nationalfonds (820.167.73).     

The group SL(2, Z s) is either perfect or prosolvable

We will show the prosolvability of SL₂ over Z and some other rings. Also we will discuss about braid groups.     

Dynamics of the continued fraction map and the spectral theory of SL (2, Z)

Oblatum 15-V-1992 & 2-II-1993     

The Selberg trace formula for SL(3,Z)

In this article the first step toward the generalization of the Selberg trace formula to the case of a rank 2 symmetric space S and a discrete group for which the fundamental region \S goes to infinity nontrivially appears. For S we use the space SL(3,)/SO(3) and for we use SL(3,). The fundamental results are Theorems 9 and 10, in which is calculated the contribution to the matrix trace of the operator K which appears in the right side of the trace formula of the expression h()d^c(), where ^c() is the continuous part of the spectral measure of the quasiregular representation on the space IL₂(\S).Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 63, pp. 8–66, 1976.     

CRITERION FOR SL（2,Z）－MATRIX TO BE CONJUGATE TO ITS INVERSE

This paper gives a necessary and sufficient condition for a matrix in SL(2, Z) to be conjugate to its inverse. This condition reduces the determination of the conjugation to solving some indeterminate equation of second degree. It yields an algorithm to determine this conjugation in finite steps based on the elementary number theory.     

Orthogonality of natural sheaves on moduli stacks of SL(2)-bundles with connections on \Bbb P^1 minus 4 points

A special kind of SL(2)-bundles with connections on \Bbb P¹\{x₁,...,x₄}\Bbb P^1\setminus\{x_1,\dots,x_4\} is considered. We construct an equivalence between the derived category of quasicoherent sheaves on the moduli stack of such bundles and the derived category of modules over a TDO ring on some (non-separated) curve.     

A separation property of the zeros of Eisenstein series for SL(2, Z)

Let E_k(z) be the Eisenstein series with weight k for the modulargroup SL(2, ). We prove that the zeros of E_k(eⁱ) interlace withthe zeros of E_k+12(eⁱ) on /2 < < 2/3. That is, any zeroof E_k(eⁱ) lies between two consecutive zeros of E_k+12(eⁱ) on/2 < < 2/3.     

On certain Dirichlet series associated with automorphic forms on SL(2,C)

In this note, we will discuss the analogy of some results of Asai [2] in the case of the automorphic forms on SL(2,C). Being combined with the base change lifting to the imaginary quadratic field, we will discuss the L-function L(s,f,Sym²) for an elliptic modular form f. The base change lifting to the field of higher degree will also be discussed.The author would like to express his hearty thanks to Prof. Nobushige Kurokawa for his valuable suggestions on the meromorphy of Euler products.     

On the spectral gap for infinite index “congruence” subgroups of SL2(Z)

A celebrated theorem of Selberg states that for congruence subgroups of SL₂(Z) there are no exceptional eigenvalues below 3/16. Extending the work of Sarnak and Xue for cocompact arithmetic lattices, we prove a generalization of Selberg’s theorem for infinite index “congruence” subgroups of SL₂(Z). For such subgroups with a high enough Hausdorff dimension of the limit set we establish a spectral gap property and consequently solve a problem of Lubotzky pertaining to expander graphs. The author was supported in part by the NSF graduate fellowship.     

Normal forms and invariants for 2-dimensional almost-Riemannian structures

2-Dimensional almost-Riemannian structures are generalized Riemannian structures on surfaces for which a local orthonormal frame is given by a Lie bracket generating pair of vector fields that can become collinear. Generically, there are three types of points: Riemannian points where the two vector fields are linearly independent, Grushin points where the two vector fields are collinear but their Lie bracket is not, and tangency points where the two vector fields and their Lie bracket are collinear and the missing direction is obtained with one more bracket.In this paper we consider the problem of finding normal forms and functional invariants at each type of point. We also require that functional invariants are “complete” in the sense that they permit to recognize locally isometric structures.The problem happens to be equivalent to the one of finding a smooth canonical parameterized curve passing through the point and being transversal to the distribution.For Riemannian points such that the gradient of the Gaussian curvature K is different from zero, we use the level set of K as support of the parameterized curve. For Riemannian points such that the gradient of the curvature vanishes (and under additional generic conditions), we use a curve which is found by looking for crests and valleys of the curvature. For Grushin points we use the set where the vector fields are parallel.Tangency points are the most complicated to deal with. The cut locus from the tangency point is not a good candidate as canonical parameterized curve since it is known to be non-smooth. Thus, we analyse the cut locus from the singular set and we prove that it is not smooth either. A good candidate appears to be a curve which is found by looking for crests and valleys of the Gaussian curvature. We prove that the support of such a curve is uniquely determined and has a canonical parametrization.