An exact trace formula and zeta functions for an infinite quantum graph with a non-standard Weyl asymptotics

This paper is a report on work in progress rather than a description of theorems which have attained their final form. The results I shall describe are part of an attempt to continue to higher dimensions the study of the relation between the Hasse-Weil zeta-functions of Shimura varieties and the Euler products associated to automorphic forms, which was initiated by Eichler, and extensively developed by Shimura for the varieties of dimension one bearing his name. The method used has its origins in an idea of Sato, which was exploited by Ihara for the Shimura varieties associated to GL(2).

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Stable Conjugacy: Definitions and Lemmas

Canadian Journal of Mathematics ◽

10.4153/cjm-1979-069-2 ◽

1979 ◽

Vol 31 (4) ◽

pp. 700-725 ◽

Cited By ~ 25

Author(s):

R. P. Langlands

Keyword(s):

Trace Formula ◽

Characteristic Zero ◽

Present Note ◽

Zeta Functions ◽

Base Change ◽

Shimura Varieties ◽

Reductive Groups ◽

Typical Problem

The purpose of the present note is to introduce some notions useful for applications of the trace formula to the study of the principle of functoriality, including base change, and to the study of zeta-functions of Shimura varieties. In order to avoid disconcerting technical digressions I shall work with reductive groups over fields of characteristic zero, but the second assumption is only a matter of convenience, for the problems caused by inseparability are not serious.The difficulties with which the trace formula confronts us are manifold. Most of them arise from the non-compactness of the quotient and will not concern us here. Others are primarily arithmetic and occur even when the quotient is compact. To see how they arise, we consider a typical problem.

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LA VARIANTE INFINITÉSIMALE DE LA FORMULE DES TRACES DE JACQUET-RALLIS POUR LES GROUPES LINÉAIRES

Journal of the Institute of Mathematics of Jussieu ◽

10.1017/s1474748016000141 ◽

2016 ◽

Vol 17 (4) ◽

pp. 735-783 ◽

Cited By ~ 4

Author(s):

Michał Zydor

Keyword(s):

Trace Formula ◽

Haar Measure ◽

Fourier Transforms ◽

Zeta Functions ◽

Adjoint Action ◽

Linear Groups ◽

Orbital Integrals ◽

Absolute Value ◽

General Linear Groups ◽

Orbital Integral

We establish an infinitesimal version of the Jacquet-Rallis trace formula for general linear groups. Our formula is obtained by integrating a kernel truncated à la Arthur multiplied by the absolute value of the determinant to the power $s\in \mathbb{C}$. It has a geometric side which is a sum of distributions $I_{\mathfrak{o}}(s,\cdot )$ indexed by the invariants of the adjoint action of $\text{GL}_{n}(\text{F})$ on $\mathfrak{gl}_{n+1}(\text{F})$ as well as a «spectral side» consisting of the Fourier transforms of the aforementioned distributions. We prove that the distributions $I_{\mathfrak{o}}(s,\cdot )$ are invariant and depend only on the choice of the Haar measure on $\text{GL}_{n}(\mathbb{A})$. For regular semi-simple classes $\mathfrak{o}$, $I_{\mathfrak{o}}(s,\cdot )$ is a relative orbital integral of Jacquet-Rallis. For classes $\mathfrak{o}$ called relatively regular semi-simple, we express $I_{\mathfrak{o}}(s,\cdot )$ in terms of relative orbital integrals regularised by means of zeta functions.

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Weyl asymptotics for Hanoi attractors

Forum Mathematicum ◽

10.1515/forum-2015-0179 ◽

2017 ◽

Vol 29 (5) ◽

pp. 1003-1021 ◽

Cited By ~ 1

Author(s):

Patricia Alonso Ruiz ◽

Uta R. Freiberg

Keyword(s):

Asymptotic Behavior ◽

Dirichlet Form ◽

Counting Function ◽

Quantum Graph ◽

Weyl Asymptotics ◽

Self Similar ◽

Resistance Form ◽

Classical Construction ◽

Suitable Measure

AbstractThis paper studies the asymptotic behavior of the eigenvalue counting function of the Laplacian on some weakly self-similar fractals called Hanoi attractors. A resistance form is constructed and equipped with a suitable measure in order to obtain a Dirichlet form and its associated Laplacian. Hereby, the classical construction for p.c.f. self-similar fractals has to be modified by combining discrete and quantum graph methods.

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