Brownian motion on the hyperbolic plane and selberg trace formula. Representations with cohomology in the discrete numdam although it is obvious that for infinitely many tiegir the multiplicities m7i. These lecture notes provide a basic introduction to selbergs trace formula. Zeta functions and regularized determinants related to the. Furthermore, we present the selberg trace formula for the above lfunction of g. Riemann, dedekind, selberg, and ihara zetas audrey terras crm montreal 2009 more details can be found in my webpage. This work offers for the first time a simultaneous treatment of a general group with the case of gl2. A selberg trace formula for hypercomplex analytic cusp. On zeros of the derivative of the threedimensional selberg zeta function minamide, makoto, illinois journal of mathematics, 2008. 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. 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. The selberg trace formula is an equality arising from computing in two different ways the traces of operators on the hilbert space l2. Let g be a semisimple lie group and h a uniform lattice in g.
Path integrals, hyperbolic spaces and selberg trace formulae. Description this thesis focuses on refinements of selberg s sieve as well as new applications of the sieve. Selbergselberg trace formula trace formula on the universal covering tree. The eichlerselberg trace formula for levelone hecke.
An introduction jens marklof school of mathematics, university of bristol, bristol bs8 1tw, u. 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. Kottwitz 1988 used the arthur selberg trace formula to prove the weil conjecture on tamagawa numbers. Selbergs trace formula is a far reaching noncommutative generalization of the poisson summation formula. Applications of the trace formula will be given in the next two chapters. In our last publication 2h, we isolated the contribution to the trace arising from the continuous spectrum, call it.
During the last 10 years or so, mathematicians have become increasingly fascinated with the selberg trace formula. This book based on lectures given by james arthur discusses the trace formula of selberg and arthur. Selberg s trace formula on the kregular tree and applications audrey terras and dorothy wallace abstract. Gelbart published on 1996 by american mathematical soc. 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. Introduction to the selberg trace formula this is a talk about the paper h. Thus, we first have to prove this property for the set of functions we deal with. The arthur selberg trace formula is an equality between two kinds of traces.
It rederives the selberg trace formula for gl 2 by a regularised process. These are notes prepared for nine lectures given at the mathematical sciences research institute, msri, berkeley during the period januarymarch 1995. Selberg super trace formula for super riemann surfaces. 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. 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. Selbergs trace formula on the kregular tree and applications. Sum formula for kloosterman sums and fourth moment of the dedekind zetafunction over the gaussian number field bruggeman, roelof w. In this chapter we introduce the selberg trace formula, which is a natural generalization of the poisson summation formula to nonabelian groups. This paper gives a version of the hyperbolic term in the selberg trace formula for sl3, z. Selberg zeta functions over function fields sciencedirect. We write tm instead of tkm for the hecke operator on the space of cusp forms sk mg. It is based on an appendix in serge langs introduction to modular forms written by don zagier, though i also. Finiteness statements and the selberg trace formula. Zagier the eichler selberg trace formula on sliz throughout this appendix we let rr1sl 2z.
Volume one deals exclusively with the case of compact quotient space. Selberg trace formula for a certain group generated. We give a decomposition formula for the lfunction of a semiregular bipartite graph g. On the selberg trace formula in the case of compact. Brownian motion on the hyperbolic plane and selberg trace. It can also be used to prove several other special cases of langlands functoriality, such as base change, for some groups. But for the case of weight one, the similar method is not effective. The trace formula was introduced by selberg, in the. Particular attention is devoted to symmetry considerations, leading to nontrivial multiplicities of eigenvalues. An introduction to the trace formula james arthur contents foreword 3 part i. The selberg trace formula for modular correspondences.
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. On the selberg trace formula in the case of compact quotient. This paper makes this version explicit for the class of representations which. 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. The zeros and poles of the selberg zetafunction, zs, can be described in terms of spectral data of the surface. Let hz, z be a function of two variables z, zin d, and assume that h as a. 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. 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. We survey graph theoretic analogues of the selberg trace and pretrace formulas along with some applications. 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.
Bumps 2001 lecture notes spectral theory of sl2,r include a proof of the selberg trace formula. 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. Download selbergs trace formula an introduction ebook pdf. Psl3, rso3, r which is detailed enough to contain information about class numbers and regulators of totally real cubic number fields. The main topic in the latter half is the selberg trace formula.
Introductory notes on the trace formula1 citeseerx. 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. The zeta function is defined in terms of the closed geodesics of the surface. Their main purpose is to provide a comprehensive development of the trace formula for psl2,r. An approach to the selberg trace formula via the selberg zeta. The selberg trace formula is of unquestionable value for the study of automorphic forms and related objects. In general, these terms require a truncation in order to converge, which leads to an equality of truncated kernels. 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. Elliptic and parabolic conjugacy classes, and selberg superzeta functions grosche, c. 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. 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. Booker, min lee and andreas strombergsson abstract. 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.
An approach to the selberg trace formula via the selberg. Version b of the selberg trace formula springerlink. 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. The selberg trace formula for psl 2,r volume 2 dennis a. Automorphic forms and the arthurselberg trace formula. 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. 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 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. This paper surveys graph theoretic analogues of the selberg trace and pre trace formulas along with some applications.
The trace formula was introduced by selberg, in the context of a semisimple lie group gand discrete subgroup. For simplicity, assume that mis a compact riemannian manifold. Further generalisation to locally compact abelian groups is required in number theory. The emphasis is laid on arthurs trace formula for glr. In noncommutative harmonic analysis, the idea is taken even further in the selberg trace formula, but takes on a much deeper character. 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. The trace formula for sl2,z can be developed for vectorvalued functions which satisfy an automorphic condition involving a group representation n. 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. This symposium contains 30 lectures that cover the significant contribution of atle selberg in the field of mathematics. 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. Explicit form of the hyperbolic term in the selberg trace. Number theory, trace formulas and discrete groups 1st.
It also gives more insight into the origin of the various terms. We present selberg s zeta function in terms of a regularized determinant of the automorphic laplacian. This proof of the trace formula is more invariant and in some respects computationally simpler than the proofs involving truncation. 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. The arthur selberg trace formula can be used to study similar correspondences on higher rank groups. It also treats the trace formula with the example of jacquets relative formula. Lfunctions and the selberg trace formulas for semiregular. The study of the selberg trace formula in this generality was previously conducted by patterson p2 and the rst author g. A connecting link between their framework and ours is. The selberg trace formula for bordered riemann surfaces. In selberg 11, he introduced the trace formula and applied it to computations of traces of hecke operators acting on the space. The character is given by the trace of certain functions on g the simplest case is when. The selberg trace formula for compact quotient 7 2. We examine spectra of dirac operators on compact hyperbolic surfaces.
Lectures on the arthurselberg trace formula internet archive. These notes were written to help remedy this situation. In fact, selberg s original paper stated the trace formula in this generality. The selberg trace formula for psl 2,r volume 1 dennis a. Shimura varieties and the selberg trace formula 5 is to be. In mathematics, the selberg trace formula, introduced by selberg 1956, is an expression for. The elchlerselberg trace formula on sl2z throughout this appendix we let r we let f be a fundamental domain for in s. We also express the selberg zeta function as a determinant of the laplacian which is composed of both discrete and continuous spectra.
Selberg trace formula, sieves mathematics, selberg, atle. Adinkras, dessins, origami, and supersymmetry spectral triples. Pure and applied mathematics eigenvalues in riemannian. 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. 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.
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