## trace formulae, explicit formulae and number theory

Warning! This set of interlinked pages is very much under construction. If you are already familiar with the subject matter, I would appreciate any help you might provide. If you are seeking information on the subject matter, I would be interested to hear what you would ultimately like this page to contain. Thank you.

 Gutzwiller Trace Formula ___  [link]   ___ Selberg Trace Formula | | [link] | | | | [link] | | Riemann's zeta function ___  [link]   ___ Riemann-Weil Explicit Formula | | [link] | | | | [link] | | Connes'(hypothetical) trace formula ___  [link]   ___ (Deninger's) Lefschetz Trace Formulae

Additional connections may be added to this diagram in due course. The individual pages are in various states of completion.

Perhaps the first questions to be asked here are "What is a trace formula in the most general sense?" and "What is an explicit formula in the most general sense?"

Deninger states:

"'Explicit formulas' have a long tradition in analytic number theory which goes back to Riemann's famous article."

His article:

[Abstract:] "We interpret the 'explicit formulas' in the sense of analytic number theory for the zeta function of an elliptic curve over a finite field as a transversal index theorem on a 3-dimensional laminated space."

may be of some general interest here.

Lang explains that explicit formulae in general can be characterised as

"...the sum of a certain function extended to the prime powers [being] essentially equal to a sum of its Mellin transform extended to the zeros of the (appropriate) zeta function."

A recent survey of trace formulae:

A. Uribe, "Trace formulae", from Quantization, the Segal-Bargmann Transform and Semiclassical Analysis (Proceedings of the First Summer School in Analysis and Mathematical Physics), Eds. S. Perez-Esteva and C. Villegas-Blas, Contemporary Mathematics volume 260 (AMS, 2000) 61-90.

other trace and explicit formulae

J.-F. Burnol appears to be onto something quite interesting, relating the Riemann zeta function to QFT, etc.:

"I give a new derivation of the Explicit Formula for the general number field K, which treats all primes in exactly the same way, whether they are discrete or archimedean, and also ramified or not. In another token, I advance a probabilistic interpretation of Weil's positivity criterion, as opposed to the usual geometrical analogies or goals. But in the end, I argue that the new formulation of the Explicit Formula signals a specific link with Quantum Fields, as opposed to the Hilbert–Pólya operator idea (which leads rather to Quantum Mechanics)."

J.-F. Burnol, "Scattering on the p-adic field and a trace formula", Int. Math. Res. Not. 2 (2000) 57-70.

J.-F. Burnol, "The Explicit Formula and the conductor operator"

J. Sjoestrand, M. Zworski, "Quantum monodromy and semi-classial trace formulae"

[abstract:] "Trace formulae provide one of the most elegant descriptions of the classical-quantum correspondence. One side of a formula is given by a trace of a quantum object, typically derived from a quantum Hamilitonian, and the other side is described in terms of closed orbits of the corresponding classical Hamiltonian. In algebraic situations, such as the original Selberg trace formula, the identities are exact, while in general they hold only in semi-classical or high-energy limits. We refer to a recent survey for an introduction and references.

In this paper we present an intermediate trace formula in which the original trace is expressed in terms of traces of quantum monodromy operators directly related to the classical dynamics. The usual trace formulae follow and in addition this approach allows handling effective Hamiltonians."

[abstract:] "We deduce from Arthur's trace formula a formula with only orbital integrals on the geometric side."

some books concerning trace formulae

K.E. Aubert, E. Bombieri, and D. Goldfeld (editors), Number Theory, Trace Formulas and Discrete Groups : Symposium in Honor of Atle Selberg, Oslo, Norway, July 14-21, 1987 (Academic Press, 1989)

C. Grosche, Path Integrals, Hyperbolic Spaces, and Selberg Trace Formulae (World Scientific, 1996)

"In this volume, a comprehensive review is given for path integration in two- and three-dimensional homogeneous spaces of constant curvature, including an enumeration of all coordinate systems which allow separation of variables in the Hamiltonian and in the path integral. The corresponding path integral solutions are presented as a tabulation. In addition, an overview is presented on some recent achievements in the theory of the Selberg trace formula on Riemann surfaces, its super generalization, and their use in mathematical physics and quantum chaos. 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 four-dimensional Euclidean space, and some further results derived from the Selberg (super-) trace formula."

I.V. Lerner, J.P. Keating, and D. Khmelnitskii (editors), Supersymmetry and Trace Formulae : Chaos and Disorder (Plenum, 1999)

"The 19 Institute papers address some questions concerning the similarities and differences between supersymmetric methods and those based on the trace formula in hopes of finding a window to an increased understanding of complex systems. Some of those are whether it is possible to use an analogue of the nonlinear sigma-model to describe spectral correlations in a single deterministic system by averaging over the energy or by introducing very weak disorder; whether ensemble averaging allows for the contribution of untypical realizations of the random potential; whether the approach to the random-matrix limit in ensembles of disordered systems is related to the corresponding approach of a single deterministically chaotic system; and the size of the exceptional set of strongly chaotic systems that do not exhibit random-matrix statistics. The Institute was held in Cambridge, England in September 1997."

J. Arthur, L. Clozel Simple Algebras, Base Change, and the Advanced Theory of the Trace Formula (Annals of Mathematics Studies, No. 120) (Princeton University Press)

"A general principle, discovered by Robert Langlands and named by him the "functoriality principle," predicts relations between automorphic forms on arithmetic subgroups of different reductive groups. Langlands functoriality relates the eigenvalues of Hecke operators acting on the automorphic forms on two groups (or the local factors of the "automorphic representations" generated by them). In the few instances where such relations have been probed, they have led to deep arithmetic consequences.

This book studies one of the simplest general problems in the theory, that of relating automorphic forms on arithmetic subgroups of GL(n,E) and GL(n,F) when E/F is a cyclic extension of number fields. (This is known as the base change problem for GL(n).) The problem is attacked and solved by means of the trace formula. The book relies on deep and technical results obtained by several authors during the last twenty years. It could not serve as an introduction to them, but, by giving complete references to the published literature, the authors have made the work useful to a reader who does not know all the aspects of the theory of automorphic forms."

G. Laumon, Cohomology of Drinfeld Modular Varieties : Automorphic Forms, Trace Formulas and Langlands Correspondence (Cambridge Studies in Advanced Mathematics) (Cambridge University Press, 1997)

"Cohomology of Drinfeld Modular Varieties provides an introduction, in two volumes, both to this subject and to the Langlands correspondence for function fields. It is based on courses given by the author who, to keep the presentation as accessible as possible, considers the simpler case of function rather than number fields; nevertheless, many important features can still be illustrated. Several appendices on background material make this a self-contained book. It will be welcomed by workers in algebraic number theory and representation theory."

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