GLOBAL ZETA FUNCTIONS OF FREUDENTHAL QUARTICS

2002 ◽  
Vol 13 (08) ◽  
pp. 797-820
Author(s):  
HIROSHI SAITO
Keyword(s):  
Vector Space ◽  
Zeta Function ◽  
Explicit Formula ◽  
Explicit Form ◽  
Zeta Functions ◽  
The Other ◽  
Vector Spaces ◽  
Prehomogeneous Vector Space ◽  
Prehomogeneous Vector Spaces

We give two applications of an explicit formula for global zeta functions of prehomogeneous vector spaces in Math. Ann.315 (1999), 587–615. One is concerned with an explicit form of global zeta functions associated with Freudenthal quartics, and the other the comparison of the zeta function of a unsaturated prehomogeneous vector space with that of the saturated one obtained from it.

scholarly journals On the convergence of the zeta function for certain prehomogeneous vector spaces

1995 ◽  
Vol 140 ◽  
pp. 1-31 ◽  
Author(s):  
Akihiko Yukie
Keyword(s):  
Invariant Measure ◽  
Zeta Function ◽  
Complex Variable ◽  
Vector Spaces ◽  
Invariant Polynomial ◽  
Connected Component ◽  
Relative Invariant ◽  
Prehomogeneous Vector Space ◽  
Prehomogeneous Vector Spaces ◽  
Rational Character

Let (G, V) be an irreducible prehomogeneous vector space defined over a number field k, P ∈ k[V] a relative invariant polynomial, and χ a rational character of G such that . For , let Gx be the stabilizer of x, and the connected component of 1 of Gx. We define L0 to be the set of such that does not have a non-trivial rational character. Then we define the zeta function for (G, Y) by the following integralwhere Φ is a Schwartz-Bruhat function, s is a complex variable, and dg” is an invariant measure.

scholarly journals A classification of irreducible prehomogeneous vector spaces and their relative invariants

1977 ◽  
Vol 65 ◽  
pp. 1-155 ◽  
Author(s):  
M. Sato ◽  
T. Kimura
Keyword(s):  
Vector Space ◽  
Linear Algebraic Group ◽  
Vector Spaces ◽  
Rational Representation ◽  
Finite Dimensional Vector Space ◽  
Prehomogeneous Vector Space ◽  
Finite Dimensional ◽  
Prehomogeneous Vector Spaces ◽  
Relative Invariants

LetGbe a connected linear algebraic group, andpa rational representation ofGon a finite-dimensional vector spaceV, all defined over the complex number fieldC.We call such a triplet (G, p, V) aprehomogeneous vector spaceifVhas a Zariski-denseG-orbit. The main purpose of this paper is to classify all prehomogeneous vector spaces whenpis irreducible, and to investigate their relative invariants and the regularity.

scholarly journals On zeta functions associated with quadratic forms of variable coefficients

1979 ◽  
Vol 73 ◽  
pp. 117-147 ◽  
Author(s):  
Toshiaki Suzuki
Keyword(s):  
General Theory ◽  
Quadratic Forms ◽  
Zeta Functions ◽  
Variable Coefficients ◽  
The Other ◽  
Class Numbers ◽  
Vector Spaces ◽  
Prehomogeneous Vector Spaces ◽  
Indefinite Quadratic ◽  
Cubic Forms

In 1938, C. L. Siegel studied zeta functions of indefinite quadratic forms ([6], c). On the other hand, M. Sato and T. Shintani constructed the general theory of zeta functions of one complex variable associated with prehomogeneous vector spaces in 1974 ([1]). Moreover T. Shintani studied several zeta functions of prehomogeneous vector spaces, especially, “Dirichlet series whose coefficients are class-numbers of integral binary cubic forms” ([3]) and “Zeta functions associated with the vector space of quadratic forms” ([2]).

scholarly journals Second variation of Selberg zeta functions and curvature asymptotics

2019 ◽  
Vol 57 (1) ◽  
pp. 23-60
Author(s):  
Ksenia Fedosova ◽  
Julie Rowlett ◽  
Genkai Zhang
Keyword(s):  
Asymptotic Behavior ◽  
Vector Bundle ◽  
Zeta Function ◽  
Explicit Formula ◽  
Zeta Functions ◽  
Teichmüller Space ◽  
Teichmuller Space ◽  
Second Variation ◽  
Selberg Zeta Function ◽  
The Second Variation

Abstract We give an explicit formula for the second variation of the logarithm of the Selberg zeta function, Z(s), on Teichmüller space. We then use this formula to determine the asymptotic behavior as $$\mathfrak {R}s \rightarrow \infty $$Rs→∞ of the second variation. As a consequence, for $$m \in {\mathbb {N}}$$m∈N, we obtain the complete expansion in m of the curvature of the vector bundle $$H^0(X_t, {\mathcal {K}}_t)\rightarrow t\in {\mathcal {T}}$$H0(Xt,Kt)→t∈T of holomorphic m-differentials over the Teichmüller space $${\mathcal {T}}$$T, for m large. Moreover, we show that this curvature agrees with the Quillen curvature up to a term of exponential decay, $$O(m^2 \mathrm{e}^{-l_0 m}),$$O(m2e-l0m), where $$l_0$$l0 is the length of the shortest closed hyperbolic geodesic.

scholarly journals Analysis and combinatorics of partition zeta functions

2020 ◽  
pp. 1-10
Author(s):  
Robert Schneider ◽  
Andrew V. Sills
Keyword(s):  
Zeta Function ◽  
Explicit Formula ◽  
Zeta Functions ◽  
Complete Information ◽  
Combinatorial Proof ◽  
Fixed Length ◽  
The Family ◽  
The Riemann Zeta Function ◽  
Riemann Zeta ◽  
Partial Fraction Decomposition

We examine “partition zeta functions” analogous to the Riemann zeta function but summed over subsets of integer partitions. We prove an explicit formula for a family of partition zeta functions already shown to have nice properties — those summed over partitions of fixed length — which yields complete information about analytic continuation, poles and trivial roots of the zeta functions in the family. Then we present a combinatorial proof of the explicit formula, which shows it to be a zeta function analog of MacMahon’s partial fraction decomposition of the generating function for partitions of fixed length.

On Zeta Functions Associated with Prehomogeneous Vector Spaces

1974 ◽  
Vol 100 (1) ◽  
pp. 131 ◽  
Author(s):  
Mikio Sato ◽  
Takuro Shintani
Keyword(s):  
Zeta Functions ◽  
Vector Spaces ◽  
Prehomogeneous Vector Spaces
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