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.