Nagoya Mathematical Journal

Hilbert-Asai Eisenstein series, regularized products, and heat kernels

Jay Jorgenson and Serge Lang

Full-text: Open access


In a famous paper, Asai indicated how to develop a theory of Eisenstein series for arbitrary number fields, using hyperbolic 3-space to take care of the complex places. Unfortunately he limited himself to class number 1. The present paper gives a detailed exposition of the general case, to be used for many applications. First, it is shown that the Eisenstein series satisfy the authors' definition of regularized products satisfying the generalized Lerch formula, and the basic axioms which allow the systematic development of the authors' theory, including the Cramér theorem. It is indicated how previous results of Efrat and Zograf for the strict Hilbert modular case extend to arbitrary number fields, for instance a spectral decomposition of the heat kernel periodized with respect to ${\rm SL}_2$ of the integers of the number field. This gives rise to a theta inversion formula, to which the authors' Gauss transform can be applied. In addition, the Eisenstein series can be twisted with the heat kernel, thus encoding an infinite amount of spectral information in one item coming from heat Eisenstein series. The main expected spectral formula is stated, but a complete exposition would require a substantial amount of space, and is currently under consideration.

Article information

Nagoya Math. J., Volume 153 (1999), 155-188.

First available in Project Euclid: 27 April 2005

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11M36: Selberg zeta functions and regularized determinants; applications to spectral theory, Dirichlet series, Eisenstein series, etc. Explicit formulas
Secondary: 11F72: Spectral theory; Selberg trace formula


Jorgenson, Jay; Lang, Serge. Hilbert-Asai Eisenstein series, regularized products, and heat kernels. Nagoya Math. J. 153 (1999), 155--188.

Export citation


  • T. Asai, On a certain function analogous to (\log \eta(z)), Nagoya Math. J., 40 (1970), 193–211.
  • I. Efrat, The Selberg trace formula for (PSL_2(R)^n) , Memoir AMS, 359 (1987).
  • I. Efrat and P. Sarnak, The determinant of the Eisenstein matrix and Hilbert class fields , Trans. AMS, 290 (1985), 815–824.
  • J. Elstrodt, E. Grunewald and J. Mennicke, Eisenstein series on three dimensional hyperbolic spaces and imaginary quadratic fields , J. reine angew. Math., 360 (1985), 160–213.
  • J. Elstrodt, E. Grunewald and J. Mennicke, Zeta functions of binary hermitian forms and special vales of Eisenstein series on three-dimensional hyperbolic space , Math. Ann., 277 (1987), 655–708.
  • R. Gangolli, On the length spectra of some compact manifolds of negative curvature , Acta Math., 121 (1986), 151–192.
  • G. Van der Geer, Hilbert modular surfaces, Springer Verlag (1980).
  • J. Jorgenson and S. Lang, Basic analysis of regularized series and products, Springer Lecture Notes 1564 (1993).
  • J. Jorgenson and S. Lang, On Cramér's theorem for general Euler products with functional equations , Math. Ann., 297 (1994), 383–416.
  • J. Jorgenson and S. Lang, Explicit formulas for regularized products and series, Springer Lecture Notes 1593 (1994).
  • J. Jorgenson and S. Lang, Extension of analytic number theory and the theory of regularized harmonic series from Dirichlet series to Bessel series , Math. Ann., 306 (1996), 75–124.
  • T. Kubota, Über diskontinuierlicher Gruppen Picardschen Typus und zuge- hörige Eisensteinsche Reihen , Nagoya Math. J., 32 (1968), 259–271.
  • S. Lang, Elliptic Functions, Addison Wesley (1973, (second edition Springer Verlag, 1987)).
  • S. Lang, Algebraic Number Theory, Addison Wesley (1970, (second edition Springer Verlag, 1994)).
  • P. Sarnak, The arithmetic and geometry of some hyperbolic three manifolds , Acta. Math., 151 (1983), 253–295.
  • C. L. Siegel, Advanced Analytic Number Theory, Lecture Notes Tata Institute (1961, (reprinted in book form, 1980)).
  • J. Szmidt, The Selberg trace formula for the Picard group (SL(2, Z[i])) , Acta Arith., 42 (1983), 291–424.
  • P. Zograf, Selberg trace formula for the Hilbert modular group of a real quadratic number field , J. Soviet Math., 19 (1982), 1637–1652.