Zeta integral

introduction

• many zeta integrals in the theory of automorphic forms can be produced or explained by appropriate choices of a Schwartz space of test functions on a spherical homogeneous space, which are in turn dictated by the geometry of affine spherical embeddings

local zeta integral

• quasicharacter on $F_v^{\times}$ are of the form $\omega_s(x)=\omega(x)|x|^s$ where $\omega$ is unitary
• $\omega$ : unitary, $s\in \mathbb{C}$
• the following converges for $\Re(s)>0$

$$ \zeta(f,\omega,s)=\int_{F_v^{\times}}f(x)\omega(x)|x|^s\, d^{\times}x $$

• analytic continuation of $Z(f,\omega,s)$
• functional equation

global zeta integral

Riemann zeta function

• $f\in \mathcal{S}(\mathbb{A})$
• define

$$ \zeta(f,s)=\int_{\mathbb{A}^{\times}}f(x)|x|^s\, d^{\times}x $$

thm

The integral converges locally uniformly for $\Re(s)>1$ and so it defines a holomorphic function in that range, which extends to an meromorphic function on $\mathbb{C}$. This function is holomorphic away from the points $s=0,1$, where it has at most simple poles of residue $-f(0)$ and $\hat{f}(0)$, respectively. The zeta integral satisfies the functional equation One has $$ \zeta(f,s)=\zeta(\widehat{f},1-s) $$

Dirichlet L-functions

• $f\in \mathcal{S}(\mathbb{A})$
• $\chi$ : character of $\mathbb{A}^{\times}/\mathbb{Q}^{\times}$ with finite image
• define

$$ \zeta(f,\chi,s)=\int_{\mathbb{A}^{\times}}f(x)\chi(x)|x|^s\, d^{\times}x $$

thm

Let $\chi\neq 1$. The integral converges locally uniformly for $\Re(s)>1$ and so it defines a holomorphic function in that range, which extends to an entire function on $\mathbb{C}$. One has $$ \zeta(f,\chi,s)=\zeta(\widehat{f},\overline{\chi},1-s) $$

articles

• http://arxiv.org/abs/1509.04835
• Li, Wen-Wei. “Zeta Integrals, Schwartz Spaces and Local Functional Equations.” arXiv:1508.05594 [math], August 23, 2015. http://arxiv.org/abs/1508.05594.