Zeta integral - 편집 역사

2020년 11월 16일 (월) 18:05에 Pythagoras0님의 편집

2020-11-16T18:05:35Z

2020년 11월 16일 (월) 17:54에 Pythagoras0님의 편집

2020-11-16T17:54:17Z

2020년 11월 13일 (금) 16:55에 imported>Pythagoras0님의 편집

2020-11-13T16:55:02Z

2020년 11월 13일 (금) 16:55에 imported>Pythagoras0님의 편집

2020-11-13T16:55:02Z

imported>Pythagoras0: /* articles */

2015-09-17T00:57:30Z

articles

2015년 8월 25일 (화) 08:51에 imported>Pythagoras0님의 편집

2015-08-25T08:51:53Z

imported>Pythagoras0: 새 문서: ==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 follo...

2015-08-25T08:51:03Z

새 문서: ==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 follo...

새 문서

==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==
* 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]]

← 이전 판		2020년 11월 13일 (금) 16:55 판
1번째 줄:		1번째 줄:
−	~~==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.
−
−
−	~~[[분류:L-functions and L-values]]~~

← 이전 판		2015년 8월 25일 (화) 08:51 판
1번째 줄:		1번째 줄:
		+	==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==		==local zeta integral==

@@ 4번째 줄: / 4번째 줄: @@
 ==local zeta integral==
-* quasicharacter on $F_v^{\times}$ are of the form $\omega_s(x)=\omega(x)|x|^s$ where $\omega$ is unitary
+* 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
-* $\omega$ : unitary, $s\in \mathbb{C}$
+* <math>\omega</math> : unitary, <math>s\in \mathbb{C}</math>
-* the following converges for $\Re(s)>0$
+* 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 $Z(f,\omega,s)$
+* analytic continuation of <math>Z(f,\omega,s)</math>
 * functional equation
 ==global zeta integral==
 ===Riemann zeta function===
-* $f\in \mathcal{S}(\mathbb{A})$
+* <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 $\Re(s)>1$ and so it defines a holomorphic function in that range, which extends to an meromorphic function on $\mathbb{C}$.
+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 $s=0,1$, where it has at most simple poles of residue $-f(0)$ and $\hat{f}(0)$, respectively. The zeta integral satisfies the functional equation
+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===
-* $f\in \mathcal{S}(\mathbb{A})$
+* <math>f\in \mathcal{S}(\mathbb{A})</math>
-* $\chi$ : character of $\mathbb{A}^{\times}/\mathbb{Q}^{\times}$ with finite image
+* <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 $\chi\neq 1$. 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
+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==

← 이전 판	2020년 11월 16일 (월) 17:54 판
(차이 없음)

@@ 43번째 줄: / 43번째 줄: @@
 ==articles==
 * 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]]