This book contains accounts of work presented during the research conference, "Zeta Functions in Geometry," held at the Tokyo Institute of Technology in August 1990. The aim of the conference was to provide an opportunity for the discussion of recent results by geometers and number theorists on zeta functions in several different categories. The exchange of ideas produced new insights on various geometric zeta functions, as well as the classical zeta functions. The zeta functions covered here are the Selberg zeta functions, the Ihara zeta functions, spectral zeta functions, and those associated with prehomogeneous vector spaces. Accessible to graduate students with background in geometry and number theory, Zeta Functions in Geometry will prove useful for its presentation of new results and up-to-date surveys.

Published for the Mathematical Society of Japan by Kinokuniya, Tokyo, and distributed worldwide, except in Japan, by the AMS.

Readership

Graduate students and researchers with a background in geometry and number theory.

Table of Contents

• S. Zelditch -- Spectrum and geodesic flow
• T. Kimura and T. Kogiso -- On adelic zeta functions of prehomogeneous vector spaces with finitely many adelic open orbits
• L. Guillopé -- Fonctions zêta de Selberg et surfaces de géométrie finie
• M. Wakayama -- The relation between the $\eta$-invariant and the spin representation in terms of the Selberg zeta function
• A. Gyoja -- Lefschetz principle in the theory of prehomogeneous vector space
• K. Takase -- On special values of Selberg zeta functions
• C. L. Epstein -- Some exact trace formulae
• K. Feng -- Zeta function; class number and cyclotomic units of cyclotomic function fields
• B. Z. Moroz -- Scalar product of Hecke $L$-functions and its application
• T. Morita -- Billiards without boundary and their zeta functions
• T. Arakawa -- Selberg zeta functions and Jacobi forms
• N. Kurokawa -- Multiple zeta functions:An example
• S. Koyama -- Zeta functions of loop groups
• A. Fujii -- Some observations concerning the distribution of the zeros of the zeta functions (I)
• H. Yoshida -- On Hermitian forms attached to zeta functions
• A. Voros -- Spectral zeta functions
• D. A. Hejhal -- Eigenvalues of the Laplacian for Hecke triangle groups
• F. Sato -- The Maass zeta functions attached to positive definite quadratic forms