The meromorphic continuation of a zeta function of Weil and Igusa Type

AbstractWe construct adelic objects for rank two integral structures on arithmetic surfaces and develop measure and integration theory, as well as elements of harmonic analysis. Using the topological Milnor K2-delic and K1×K1-delic objects associated to an arithmetic surface, an adelic zeta integral is defined. Its unramified version is closely related to the square of the zeta function of the surface. For a proper regular model of an elliptic curve over a global field, a two-dimensional version of the theory of Tate and Iwasawa is derived. Using adelic analytic duality and a two-dimensional theta formula, the study of the zeta integral is reduced to the study of a boundary integral term. The work includes first applications to three fundamental properties of the zeta function: its meromorphic continuation and functional equation and a hypothesis on its mean periodicity; the location of its poles and a hypothesis on the permanence of the sign of the fourth logarithmic derivative of a boundary function; and its pole at the central point where the boundary integral explicitly relates the analytic and arithmetic ranks.

Download Full-text

Meromorphic continuation for the zeta function of a Dwork hypersurface

Algebra & Number Theory ◽

10.2140/ant.2010.4.839 ◽

2010 ◽

Vol 4 (7) ◽

pp. 839-854

Author(s):

Thomas Barnet-Lamb

Keyword(s):

Zeta Function ◽

Meromorphic Continuation

Download Full-text

ON EPSTEIN'S ZETA FUNCTION OF HUMBERT FORMS

International Journal of Number Theory ◽

10.1142/s1793042108001407 ◽

2008 ◽

Vol 04 (03) ◽

pp. 387-401 ◽

Cited By ~ 1

Author(s):

RENAUD COULANGEON

Keyword(s):

Functional Equation ◽

Zeta Function ◽

Numerical Integration ◽

Complex Number ◽

Quadratic Forms ◽

Zeta Functions ◽

Number Fields ◽

Positive Definite ◽

Meromorphic Continuation ◽

General Criterion

The Epstein ζ function ζ(Γ,s) of a lattice Γ is defined by a series which converges for any complex number s such that ℜ s > n/2, and admits a meromorphic continuation to the complex plane, with a simple pole at s = n/2. The question as to which Γ, for a fixed s, minimizes ζ(Γ,s), has a long history, dating back to Sobolev's work on numerical integration, and subsequent papers by Delone and Ryshkov among others. This was also investigated more recently by Sarnak and Strombergsson. The present paper is concerned with similar questions for positive definite quadratic forms over number fields, also called Humbert forms. We define Epstein zeta functions in that context and study their meromorphic continuation and functional equation, this being known in principle but somewhat hard to find in the literature. Then, we give a general criterion for a Humbert form to be finally ζ extreme, which we apply to a family of examples in the last section.

Download Full-text

The meromorphic continuation of the zeta function of Siegel modular threefolds over totally real fields

Functiones et Approximatio Commentarii Mathematici ◽

10.7169/facm/2012.47.2.1 ◽

2012 ◽

Vol 47 (2) ◽

pp. 143-148

Author(s):

Cristian Virdol

Keyword(s):

Zeta Function ◽

Meromorphic Continuation ◽

Totally Real ◽

Totally Real Fields ◽

Modular Threefolds

Download Full-text

ON WITTEN MULTIPLE ZETA-FUNCTIONS ASSOCIATED WITH SEMI-SIMPLE LIE ALGEBRAS IV

Glasgow Mathematical Journal ◽

10.1017/s0017089510000613 ◽

2010 ◽

Vol 53 (1) ◽

pp. 185-206 ◽

Cited By ~ 4

Author(s):

YASUSHI KOMORI ◽

KOHJI MATSUMOTO ◽

HIROFUMI TSUMURA

Keyword(s):

Zeta Function ◽

Lie Algebras ◽

Generating Functions ◽

Zeta Functions ◽

Bernoulli Polynomials ◽

Root Systems ◽

Previous Method ◽

Meromorphic Continuation ◽

Functional Relations ◽

Multiple Zeta

AbstractIn our previous work, we established the theory of multi-variable Witten zeta-functions, which are called the zeta-functions of root systems. We have already considered the cases of types A2, A3, B2, B3 and C3. In this paper, we consider the case of G2-type. We define certain analogues of Bernoulli polynomials of G2-type and study the generating functions of them to determine the coefficients of Witten's volume formulas of G2-type. Next, we consider the meromorphic continuation of the zeta-function of G2-type and determine its possible singularities. Finally, by using our previous method, we give explicit functional relations for them which include Witten's volume formulas.

Download Full-text