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Eisenstein series on three-dimensional hyperbolic space and imaginary quadratic number fields. (English) Zbl 0555.10012

Let K be an imaginary quadratic number field with class number h and ring of integers \({\mathfrak o}\). The group \(\Gamma =PSL(2,{\mathfrak o})\) acts discontinuously on the three-dimensional hyperbolic space \({\mathbb{H}}={\mathbb{C}}\times]0,\infty [\), and \(\Gamma\) \(\setminus {\mathbb{H}}\) has finite volume. The set \({\mathbb{P}}^ 1(K)\) of cusps of \(\Gamma\) splits into h orbits with respect to \(\Gamma\). Hence there are h linearly independent Eisenstein series for \(\Gamma\). These are conveniently parametrized by the set \({\mathfrak M}\) of all finitely generated \({\mathfrak o}\)-modules \({\mathfrak m}\neq \{0\}\) of K. For every \({\mathfrak m}\in {\mathfrak M}\) we define two Eisenstein series \(E_{{\mathfrak m}}(P,s)\), \(\hat E_{{\mathfrak m}}(P,s)\) (P\(\in {\mathbb{H}}\), Re s\(>1)\). The linear relations between the two systems of Eisenstein series are explicitly determined. The corresponding matrix involves the zeta-functions of the ideal classes of \({\mathfrak o}\), and its inverse can be given in terms of the L-functions for the characters on the group of ideal classes of \({\mathfrak o}.\)
The main result of the present work is an explicit determination of the Fourier expansion of \(\hat E_{{\mathfrak m}}(P,s)\) at an arbitrary cusp of \(\Gamma\). The zeroth Fourier coefficients are governed by zeta-functions of ideal classes of \({\mathfrak o}\), and the higher Fourier coefficients involve certain generalized divisor sums. This yields the analytic continuation and functional equation of the Eisenstein series and two versions of the Kronecker limit formula.
In some cases we can evaluate the Eisenstein series at special points \(P\in {\mathbb{H}}\) in terms of the Riemann zeta-function and other Dirichlet series. Combining these results with the Fourier expansion of the Eisenstein series we obtain several identities relating the Riemann zeta- function and various L-series with series of Bessel functions. The most intriguing numerical results arise from a comparison of the special values of the Eisenstein series with the two versions of the Kronecker limit formula.
Further applications of the Fourier expansion include the study of zeta- functions of ideal classes of \({\mathfrak o}\), the asymptotics of certain sums of divisor sums, new proofs of the non-vanishing of the L-series for \({\mathfrak o}\) for Re \(s\geq 1\) and a new proof of Humbert’s formula for the volume of \(\Gamma\) \(\setminus {\mathbb{H}}\).

MSC:

11F11 Holomorphic modular forms of integral weight
11F67 Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols
11M06 \(\zeta (s)\) and \(L(s, \chi)\)
11M35 Hurwitz and Lerch zeta functions
30F40 Kleinian groups (aspects of compact Riemann surfaces and uniformization)
11R11 Quadratic extensions
11R42 Zeta functions and \(L\)-functions of number fields
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