"Zeta integral"의 두 판 사이의 차이
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Pythagoras0 (토론 | 기여) |
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| + | ==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 <math>F_v^{\times}</math> are of the form <math>\omega_s(x)=\omega(x)|x|^s</math> where <math>\omega</math> is unitary | ||
| + | * <math>\omega</math> : unitary, <math>s\in \mathbb{C}</math> | ||
| + | * the following converges for <math>\Re(s)>0</math> | ||
| + | :<math> | ||
| + | \zeta(f,\omega,s)=\int_{F_v^{\times}}f(x)\omega(x)|x|^s\, d^{\times}x | ||
| + | </math> | ||
| + | * analytic continuation of <math>Z(f,\omega,s)</math> | ||
| + | * functional equation | ||
| + | |||
| + | ==global zeta integral== | ||
| + | ===Riemann zeta function=== | ||
| + | * <math>f\in \mathcal{S}(\mathbb{A})</math> | ||
| + | * define | ||
| + | :<math> | ||
| + | \zeta(f,s)=\int_{\mathbb{A}^{\times}}f(x)|x|^s\, d^{\times}x | ||
| + | </math> | ||
| + | ;thm | ||
| + | The integral converges locally uniformly for <math>\Re(s)>1</math> and so it defines a holomorphic function in that range, which extends to an meromorphic function on <math>\mathbb{C}</math>. | ||
| + | This function is holomorphic away from the points <math>s=0,1</math>, where it has at most simple poles of residue <math>-f(0)</math> and <math>\hat{f}(0)</math>, respectively. The zeta integral satisfies the functional equation | ||
| + | One has | ||
| + | :<math> | ||
| + | \zeta(f,s)=\zeta(\widehat{f},1-s) | ||
| + | </math> | ||
| + | |||
| + | |||
| + | ===Dirichlet L-functions=== | ||
| + | * <math>f\in \mathcal{S}(\mathbb{A})</math> | ||
| + | * <math>\chi</math> : character of <math>\mathbb{A}^{\times}/\mathbb{Q}^{\times}</math> with finite image | ||
| + | * define | ||
| + | :<math> | ||
| + | \zeta(f,\chi,s)=\int_{\mathbb{A}^{\times}}f(x)\chi(x)|x|^s\, d^{\times}x | ||
| + | </math> | ||
| + | ;thm | ||
| + | Let <math>\chi\neq 1</math>. The integral converges locally uniformly for <math>\Re(s)>1</math> and so it defines a holomorphic function in that range, which extends to an entire function on <math>\mathbb{C}</math>. One has | ||
| + | :<math> | ||
| + | \zeta(f,\chi,s)=\zeta(\widehat{f},\overline{\chi},1-s) | ||
| + | </math> | ||
| + | |||
| + | ==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. | ||
| + | |||
| + | |||
| + | [[분류:L-functions and L-values]] | ||
| + | [[분류:migrate]] | ||
2020년 11월 16일 (월) 11:05 기준 최신판
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.