Book contents
- Frontmatter
- Contents
- Preface
- Part I Finite Abelian Groups
- Part II Finite Nonabelian Groups
- 15 Fourier Transform and Representations of Finite Groups
- 16 Induced Representations
- 17 The Finite ax + b Group
- 18 The Heisenberg Group
- 19 Finite Symmetric Spaces–Finite Upper Half Plane Hq
- 20 Special Functions on Hq – K-Bessel and Spherical
- 21 The General Linear Group (Expression not displayed)
- 22 Selberg's Trace Formula and Isospectral Non-isomorphic Graphs
- 23 The Trace Formula on Finite Upper Half Planes
- 24 Trace Formula For a Tree and Ihara's Zeta Function
- References
- Index
22 - Selberg's Trace Formula and Isospectral Non-isomorphic Graphs
from Part II - Finite Nonabelian Groups
Published online by Cambridge University Press: 06 July 2010
- Frontmatter
- Contents
- Preface
- Part I Finite Abelian Groups
- Part II Finite Nonabelian Groups
- 15 Fourier Transform and Representations of Finite Groups
- 16 Induced Representations
- 17 The Finite ax + b Group
- 18 The Heisenberg Group
- 19 Finite Symmetric Spaces–Finite Upper Half Plane Hq
- 20 Special Functions on Hq – K-Bessel and Spherical
- 21 The General Linear Group (Expression not displayed)
- 22 Selberg's Trace Formula and Isospectral Non-isomorphic Graphs
- 23 The Trace Formula on Finite Upper Half Planes
- 24 Trace Formula For a Tree and Ihara's Zeta Function
- References
- Index
Summary
Selberg's theory is one of the biggest guns.
T. Tamagawa [1960]Some readers may feel that Hejhal has created a “monster.”
D. Hejhal [1992]All of this looks to be elaborate and extremely difficult, as indeed it is; so it is very helpful to understand it for simple examples.…
R. Langlands (in Aubert et al. [1989, p. 152])Selberg's trace formula was formulated in 1956 by Selberg [1989, Vol. I, pp. 423–63] for arithmetic groups Г like SL(2, ℤ) acting on the Poincaré upper half plane H defined in (2) of Chapter 23. Despite the slightly fearsome quotes above, it has proved to be extremely useful. Applications include:
a derivation of Weyl's law on the distribution of eigenvalues of the Laplacian on Г\H,
an answer to the question: Can you hear the shape of Г\H?,
the discussion of the analytic properties of Selberg's zeta function (which are essentially equivalent to the trace formula).
We will summarize some of this in Chapter 23. Our main goal here is to consider finite analogues of some of these results. At this point the reader might wish to review the discussion “beginnings of a trace formula” at the end of Chapter 3 as well as Chapter 12 (Poisson's summation formula).
Finite versions of the trace formula do not seem to have been much studied. We first began to think about these things after reading Arthur's derivation of the Frobenius reciprocity law from the trace formula in Aubert et al. [1989, pp. 11–27].
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- Fourier Analysis on Finite Groups and Applications , pp. 385 - 393Publisher: Cambridge University PressPrint publication year: 1999