On the zeta-functions of the algebraic curves uniformized by certain automorphic functions

AbstractLet X be a complete, geometrically irreducible, singular, algebraic curve defined over a field of characteristic p big enough. Given a local ring Op,x at a rational singular point P of X, we attached a universal zeta function which is a rational function and admits a functional equation if Op,x is Gorenstein. This universal zeta function specializes to other known zeta functions and Poincaré series attached to singular points of algebraic curves. In particular, for the local ring attached to a complex analytic function in two variables, our universal zeta function specializes to the generalized Poincaré series introduced by Campillo, Delgado, and Gusein-Zade.

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A transfer-operator-based relation between Laplace eigenfunctions and zeros of Selberg zeta functions

Ergodic Theory and Dynamical Systems ◽

10.1017/etds.2018.51 ◽

2018 ◽

Vol 40 (3) ◽

pp. 612-662

Author(s):

ALEXANDER ADAM ◽

ANKE POHL

Keyword(s):

Transfer Operator ◽

Zeta Functions ◽

Cusp Forms ◽

Dimensional Representation ◽

Triangle Group ◽

Selberg Zeta Function ◽

One Dimensional ◽

Transfer Operators ◽

Finite Dimensional ◽

Automorphic Functions

Over the last few years Pohl (partly jointly with coauthors) has developed dual ‘slow/fast’ transfer operator approaches to automorphic functions, resonances, and Selberg zeta functions for a certain class of hyperbolic surfaces $\unicode[STIX]{x1D6E4}\backslash \mathbb{H}$ with cusps and all finite-dimensional unitary representations $\unicode[STIX]{x1D712}$ of $\unicode[STIX]{x1D6E4}$. The eigenfunctions with eigenvalue 1 of the fast transfer operators determine the zeros of the Selberg zeta function for $(\unicode[STIX]{x1D6E4},\unicode[STIX]{x1D712})$. Further, if $\unicode[STIX]{x1D6E4}$ is cofinite and $\unicode[STIX]{x1D712}$ is the trivial one-dimensional representation then highly regular eigenfunctions with eigenvalue 1 of the slow transfer operators characterize Maass cusp forms for $\unicode[STIX]{x1D6E4}$. Conjecturally, this characterization extends to more general automorphic functions as well as to residues at resonances. In this article we study, without relying on Selberg theory, the relation between the eigenspaces of these two types of transfer operators for any Hecke triangle surface $\unicode[STIX]{x1D6E4}\backslash \mathbb{H}$ of finite or infinite area and any finite-dimensional unitary representation $\unicode[STIX]{x1D712}$ of the Hecke triangle group $\unicode[STIX]{x1D6E4}$. In particular, we provide explicit isomorphisms between relevant subspaces. This solves a conjecture by Möller and Pohl, characterizes some of the zeros of the Selberg zeta functions independently of the Selberg trace formula, and supports the previously mentioned conjectures.

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The Large Sieve, Monodromy, and Zeta Functions of Algebraic Curves, 2: Independence of the Zeros

International Mathematics Research Notices ◽

10.1093/imrn/rnn091 ◽

2010 ◽

Author(s):

E. Kowalski

Keyword(s):

Algebraic Curves ◽

Zeta Functions ◽

Large Sieve

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Motivic zeta functions for curve singularities

Nagoya Mathematical Journal ◽

10.1017/s0027763000009934 ◽

2010 ◽

Vol 198 ◽

pp. 47-75

Author(s):

J. J. Moyano-Fernández ◽

W. A. Zúňiga-Galindo

Keyword(s):

Functional Equation ◽

Zeta Function ◽

Local Ring ◽

Algebraic Curve ◽

Algebraic Curves ◽

Zeta Functions ◽

Poincaré Series ◽

Poincare Series ◽

Motivic Zeta Functions ◽

Curve Singularities

AbstractLetXbe a complete, geometrically irreducible, singular, algebraic curve defined over a field of characteristicpbig enough. Given a local ringOp,x at a rational singular pointPofX, we attached a universal zeta function which is a rational function and admits a functional equation ifOp,x is Gorenstein. This universal zeta function specializes to other known zeta functions and Poincaré series attached to singular points of algebraic curves. In particular, for the local ring attached to a complex analytic function in two variables, our universal zeta function specializes to the generalized Poincaré series introduced by Campillo, Delgado, and Gusein-Zade.

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