Zeta functions associated to cones and their special values.

*(English)*Zbl 0712.14009
Automorphic forms and geometry of arithmetic varieties, Adv. Stud. Pure Math. 15, 1-27 (1989).

[For the entire collection see Zbl 0688.00008.]

This is a survey article on the zeta-functions associated to self-dual cones but includes the authors’ recent results and conjectures.

In § 1, the authors summarize basic facts on self-dual homogeneous cones and associated \(\Gamma\)-functions. - In § 2, the authors define a set of zeta-functions following Sato-Shintani’s idea on the zeta- functions of a prehomogeneous vector space and give an explicit form of the system of functional equations. Further, under a certain condition, the authors define a new kind of L-functions as a suitable linear combination of zeta-functions as above and give an individual functional equation. New results are given in \(theorems 2.3.9\text{ and } 2.4.1\), which are concerned with residues and special values for these zeta- and L-functions.

In § 3, under a certain condition the authors study the geometric invariants of cusp singularities which appear in the compactification of arithmetic quotient for the tube domain associated with a given cone and give three conjectures which generalize the Hirzebruch conjecture in the Hilbert modular case. - In the final § 4, more generally the authors define zeta-functions attached to a Tsuchihasi singularity and give a formula for its value \(at\quad 0.\) The authors hope that their approach may be a new possibility of attacking their conjectures.

This is a survey article on the zeta-functions associated to self-dual cones but includes the authors’ recent results and conjectures.

In § 1, the authors summarize basic facts on self-dual homogeneous cones and associated \(\Gamma\)-functions. - In § 2, the authors define a set of zeta-functions following Sato-Shintani’s idea on the zeta- functions of a prehomogeneous vector space and give an explicit form of the system of functional equations. Further, under a certain condition, the authors define a new kind of L-functions as a suitable linear combination of zeta-functions as above and give an individual functional equation. New results are given in \(theorems 2.3.9\text{ and } 2.4.1\), which are concerned with residues and special values for these zeta- and L-functions.

In § 3, under a certain condition the authors study the geometric invariants of cusp singularities which appear in the compactification of arithmetic quotient for the tube domain associated with a given cone and give three conjectures which generalize the Hirzebruch conjecture in the Hilbert modular case. - In the final § 4, more generally the authors define zeta-functions attached to a Tsuchihasi singularity and give a formula for its value \(at\quad 0.\) The authors hope that their approach may be a new possibility of attacking their conjectures.

Reviewer: K.Katayama

##### MSC:

14G10 | Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture) |

14M17 | Homogeneous spaces and generalizations |

14B05 | Singularities in algebraic geometry |