Fourier Analysis on Finite Groups and Applications
We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.
We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.
Selberg's theory is one of the biggest guns.
Some readers may feel that Hejhal has created a “monster.”
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.…
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].
Email your librarian or administrator to recommend adding this book to your organisation's collection.