The main topic in the latter half is the selberg trace formula. Selbergselberg trace formula trace formula on the universal covering tree. This paper gives a version of the hyperbolic term in the selberg trace formula for sl3, z. This paper makes this version explicit for the class of representations which. The selberg trace formula for psl 2,r volume 2 dennis a. Psl3, rso3, r which is detailed enough to contain information about class numbers and regulators of totally real cubic number fields. The character is given by the trace of certain functions on g the simplest case is when. A connecting link between their framework and ours is. Earlier expositions of the gl2 trace formula, such as ge and gj, will be referred to occasionally. In this section we will assume that g is a semisimple lie group with finite center. Riemann, dedekind, selberg, and ihara zetas audrey terras crm montreal 2009 more details can be found in my webpage. These lecture notes provide a basic introduction to selbergs trace formula. The trace formula was introduced by selberg, in the context of a semisimple lie group gand discrete subgroup.
Applications of the trace formula will be given in the next two chapters. The zeta function is defined in terms of the closed geodesics of the surface. Selbergs trace formula on the kregular tree and applications. Further generalisation to locally compact abelian groups is required in number theory. The selberg trace formula for psl 2,r volume 1 dennis a. Questions of trace class find, read and cite all the research you need on researchgate we use cookies to make interactions with our website easy and meaningful, to better understand the use. We write tm instead of tkm for the hecke operator on the space of cusp forms sk mg. This paper surveys graph theoretic analogues of the selberg trace and pre trace formulas along with some applications. In mathematics, the selberg trace formula, introduced by selberg 1956, is an expression for. Brownian motion on the hyperbolic plane and selberg trace. It is based on an appendix in serge langs introduction to modular forms written by don zagier, though i also. In mathematics, the selberg trace formula, introduced by selberg 1956, is an expression for the character of the unitary representation of g on the space l 2 g. It also gives more insight into the origin of the various terms.
The arthur selberg trace formula can be used to study similar correspondences on higher rank groups. Number theory, trace formulas and discrete groups 1st. The selberg trace formula is an equality arising from computing in two different ways the traces of operators on the hilbert space l2. The trace formula is currently the only avail able tool to analyze the fine structure of the spectrum of no individual formulas for its eigenvalues are known. For a general fuchsian group of the first kind with an arbitrary unitary representation we define zeta functions related to the contributions of the identity, hyperbolic, elliptic and parabolic conjugacy classes in selberg s trace formula. The selberg trace formula for bordered riemann surfaces. Explicit form of the hyperbolic term in the selberg trace. An essential ingredient of the proof of the selberg trace formula is the finiteness of dimension of the clifford right module of the set of cusp forms. This symposium contains 30 lectures that cover the significant contribution of atle selberg in the field of mathematics. The zeros and poles of the selberg zetafunction, zs, can be described in terms of spectral data of the surface. Zagier the eichler selberg trace formula on sliz throughout this appendix we let rr1sl 2z. It also treats the trace formula with the example of jacquets relative formula. It rederives the selberg trace formula for gl 2 by a regularised process.
The basic idea is to compute the trace of the iterated resolvent kernel of the hyperbolic laplacian in order to arrive at the logarithmic derivative of the selberg zetafunction. We examine spectra of dirac operators on compact hyperbolic surfaces. It can also be used to prove several other special cases of langlands functoriality, such as base change, for some groups. The simple version of the trace formula, derived by arthur from his general trace formula for appropriate choices of data, can also be stabilized, and the. An introduction to the trace formula james arthur contents foreword 3 part i. Selberg zeta functions over function fields sciencedirect. Gelbart published on 1996 by american mathematical soc.
The volume also contains results on the study of the properties of a particular integrable billiard system in the hyperbolic plane, a proposal concerning interbasis expansions for spheroidal coordinate systems in fourdimensional euclidean space, and some further results derived from the selberg super trace formula. In fact, selberg s original paper stated the trace formula in this generality. In our last publication 2h, we isolated the contribution to the trace arising from the continuous spectrum, call it. Zeta functions and regularized determinants related to the. In general, these terms require a truncation in order to converge, which leads to an equality of truncated kernels. The trace formula for sl2,z can be developed for vectorvalued functions which satisfy an automorphic condition involving a group representation n. On the selberg trace formula in the case of compact. We survey graph theoretic analogues of the selberg trace and pretrace formulas along with some applications.
In this chapter we introduce the selberg trace formula, which is a natural generalization of the poisson summation formula to nonabelian groups. Pdf format is widely accepted and good for printing. The purpose of these notes is to describe the contents of arthurs earlier, foundational papers on the trace formula. Their main purpose is to provide a comprehensive development of the trace formula for psl2,r. Thus, we first have to prove this property for the set of functions we deal with. But for the case of weight one, the similar method is not effective. The emphasis is laid on arthurs trace formula for glr.
The arthur selberg trace formula is an equality between two kinds of traces. The selberg trace formula for modular correspondences. Let hz, z be a function of two variables z, zin d, and assume that h as a. A selberg trace formula is derived for the laplacebeltrami operator on bordered riemann surfaces with dirichlet or neumann boundary conditions, respectively, using a construction via the compact double of the surface, for which the standard trace formula is valid. Lfunctions and the selberg trace formulas for semiregular. The selberg trace formula for groups of frank one introduction an important tool for the study of automorphic forms is a non abelian analogue of the poisson summation formula, generally known as the selberg trace formula. The eichler selberg trace formula for levelone hecke operators alex barron may 20, 20 1 introduction this paper explains the steps involved in the proof of the eichler selberg trace formula for hecke operators of level one. Particular attention is devoted to symmetry considerations, leading to nontrivial multiplicities of eigenvalues. Elliptic and parabolic conjugacy classes, and selberg superzeta functions grosche, c. Lectures on the arthurselberg trace formula internet archive. Selberg s trace formula on the kregular tree and applications audrey terras and dorothy wallace abstract. Pure and applied mathematics eigenvalues in riemannian. These are notes prepared for nine lectures given at the mathematical sciences research institute, msri, berkeley during the period januarymarch 1995.
An approach to the selberg trace formula via the selberg. Selbergs trace formula is a far reaching noncommutative generalization of the poisson summation formula. Selberg super trace formula for super riemann surfaces. A detailed discussion of the spherical functions is given. In addition, an overview is presented on some recent achievements in the theory of the selberg trace formula on riemann surfaces, its super generalization, their use in mathematical physics and string theory, and some further results derived from the selberg super trace formula. An approach to the selberg trace formula via the selberg zeta. The major tool we will use is the selberg trace formula, which states the trace of a certain compact selfadjoint operator on l2s can be expressed as a sum over conjugacy. On zeros of the derivative of the threedimensional selberg zeta function minamide, makoto, illinois journal of mathematics, 2008. The selberg trace formula is of unquestionable value for the study of automorphic forms and related objects. Introductory notes on the trace formula1 citeseerx. We fix a weight k even we write 71m instead of tkm for the hecke operator on the space of cusp forms m let hz, z be a function of two variables z, z in s, and assume that h as a. Our main result is a selberg trace formula for dirac operators on hyperbolic surfaces.
Bumps 2001 lecture notes spectral theory of sl2,r include a proof of the selberg trace formula. Symposium in honor of atle selberg oslo, norway, july 1421, 1987 is a collection of papers presented at the 1987 selberg symposium, held at the university of oslo. We give a decomposition formula for the lfunction of a semiregular bipartite graph g. Sum formula for kloosterman sums and fourth moment of the dedekind zetafunction over the gaussian number field bruggeman, roelof w. We also express the selberg zeta function as a determinant of the laplacian which is composed of both discrete and continuous spectra. For any hyperbolic surface of finite area there is an associated selberg zetafunction. The elchlerselberg trace formula on sl2z throughout this appendix we let r we let f be a fundamental domain for in s. In selberg 11, he introduced the trace formula and applied it to computations of traces of hecke operators acting on the space of cusp forms of weight greater than or equal to two. Introduction to the selberg trace formula this is a talk about the paper h.
Version b of the selberg trace formula springerlink. Letting 1 be a given discrete hyperbolic subgroup of psl 2 r, we will consider the compact smooth riemann surface m1h2 with genus larger than or equal to two and will study the selberg trace formula for the semigroup generated by h b acting on automorphic forms with respect to. The eichlerselberg trace formula for levelone hecke. Automorphic forms and the arthurselberg trace formula. Selberg trace formula, sieves mathematics, selberg, atle. The trace formula was introduced by selberg, in the. We compute explicitly the selberg trace formula for principal congruence subgroups of pgl2, f q t which is the modular group in positive characteristic cases. Kottwitz 1988 used the arthur selberg trace formula to prove the weil conjecture on tamagawa numbers. Download selbergs trace formula an introduction ebook pdf. In this talk we will rst discuss poisson summation formula and try to understand how selberg trace formula is a generalisation of poisson summation formula in the case where g is locally compact group and is. In mathematics the selberg trace formula introduced by selberg 1956 is an expression for the character of the unitary representation of g on the space l 2 g of square integrable functions where g is a lie group and a cofinite discrete groupthe character. In noncommutative harmonic analysis, the idea is taken even further in the selberg trace formula, but takes on a much deeper character. The basic idea is to compute the trace of the iterated resolvent kernel of the hyperbolic laplacian in order to arrive at the logarithmic derivative of the. During the last 10 years or so, mathematicians have become increasingly fascinated with the selberg trace formula.
Description this thesis focuses on refinements of selberg s sieve as well as new applications of the sieve. Furthermore, we present the selberg trace formula for the above lfunction of g. Selbergs trace for mula as applied to a compact riemann surface 1972. The study of the selberg trace formula in this generality was previously conducted by patterson p2 and the rst author g. Volume one deals exclusively with the case of compact quotient space. Atle selberg, harmonic analysis and discontinuous groups in weakly symmetric riemannian spaces with applications to dirichlet series, journal of the indian mathematical society 20 1956 4787. Selbergs trace formula on the kregular tree and applications audrey terras and dorothy wallace received 7 november 2001 we survey graph theoretic analogues of the selberg trace and pretrace formulas along with some applications. This proof of the trace formula is more invariant and in some respects computationally simpler than the proofs involving truncation.
Selberg trace formula for a certain group generated. Shimura varieties and the selberg trace formula 5 is to be. There is an obvious similarity between our results for the l2index and the index formulae of atiyahpatodisinger 3 for elliptic problems with global boundary conditions. A selberg trace formula for hypercomplex analytic cusp.
Let g be a semisimple lie group and h a uniform lattice in g. These notes were written to help remedy this situation. This work offers for the first time a simultaneous treatment of a general group with the case of gl2. Path integrals, hyperbolic spaces and selberg trace formulae. Representations with cohomology in the discrete numdam although it is obvious that for infinitely many tiegir the multiplicities m7i. This book based on lectures given by james arthur discusses the trace formula of selberg and arthur. Brownian motion on the hyperbolic plane and selberg trace formula.
Finiteness statements and the selberg trace formula. The selberg trace formula for psl2,r, volume i springer lecture notes 548, 1976 the tamagawa paper is important because it is the origin of the representation theoretic approach to the trace formula. In selberg 11, he introduced the trace formula and applied it to computations of traces of hecke operators acting on the space. We derive a fully explicit version of the selberg trace formula for twistminimal maass forms of weight 0 and arbitrary conductor and nebentypus character, and apply it to prove two theorems. For simplicity, assume that mis a compact riemannian manifold. We present selberg s zeta function in terms of a regularized determinant of the automorphic laplacian.