Hecke L-functions - 편집 역사

2021년 2월 17일 (수) 10:08에 Pythagoras0님의 편집

2021-02-17T10:08:34Z

Pythagoras0: /* 메타데이터 */ 새 문단

2020-12-28T08:24:37Z

메타데이터: 새 문단

2020년 11월 16일 (월) 12:27에 Pythagoras0님의 편집

2020-11-16T12:27:04Z

2020년 11월 13일 (금) 09:35에 imported>Pythagoras0님의 편집

2020-11-13T09:35:01Z

imported>Pythagoras0: section 'articles' updated

2016-06-02T08:18:59Z

section 'articles' updated

imported>Pythagoras0: /* articles */

2015-10-29T11:45:38Z

articles

2015년 8월 25일 (화) 08:50에 imported>Pythagoras0님의 편집

2015-08-25T08:50:03Z

imported>Pythagoras0: /* expositions */

2015-02-20T08:27:55Z

expositions

imported>Pythagoras0: /* expositions */

2015-01-02T09:28:27Z

expositions

2015년 1월 2일 (금) 09:10에 imported>Pythagoras0님의 편집

2015-01-02T09:10:15Z

@@ 94번째 줄: / 94번째 줄: @@
 [[분류:migrate]]
-== 메타데이터 ==
+==메타데이터==
 ===위키데이터===
 * ID :  [https://www.wikidata.org/wiki/Q1948965 Q1948965]

← 이전 판		2020년 12월 28일 (월) 08:24 판
93번째 줄:		93번째 줄:
	[[분류:L-functions and L-values]]		[[분류:L-functions and L-values]]
	[[분류:migrate]]		[[분류:migrate]]
		+
		+	== 메타데이터 ==
		+
		+	===위키데이터===
		+	* ID : [https://www.wikidata.org/wiki/Q1948965 Q1948965]

@@ 5번째 줄: / 5번째 줄: @@
 * In 1945, Artin and whaples defined the adele ring of an algebraic number field
 * In 1950, Tate carried out the suggestion of Artin to use harmonic analysis of adele groups to prove Hecke's theorems abour L-functions attached to grossencharacters
-** for example, analytic continuation of classical $\zeta$-functions and Dirichlet $L$-functions
+** for example, analytic continuation of classical <math>\zeta</math>-functions and Dirichlet <math>L</math>-functions
 * the following is taken from '''[Leahy2010]'''
 <blockquote>
@@ 21번째 줄: / 21번째 줄: @@
 ==overview==
-* A Hecke character (or Größencharakter) of a number field $K$ is defined to be a quasicharacter of the idèle class group of $K$
+* A Hecke character (or Größencharakter) of a number field <math>K</math> is defined to be a quasicharacter of the idèle class group of <math>K</math>
-* Robert Langlands interpreted Hecke characters as automorphic forms on the reductive algebraic group GL(1) over the ring of adeles of $K$
+* Robert Langlands interpreted Hecke characters as automorphic forms on the reductive algebraic group GL(1) over the ring of adeles of <math>K</math>
-* Let $E⁄K$ be an abelian Galois extension with Galois group $G$
+* Let <math>E⁄K</math> be an abelian Galois extension with Galois group <math>G</math>
-* Then for any character $\sigma:G \to \mathbb{C}^{\times}$ (i.e. one-dimensional complex representation of the group $G$), there exists a Hecke character $\chi$ of $K$ such that
+* Then for any character <math>\sigma:G \to \mathbb{C}^{\times}</math> (i.e. one-dimensional complex representation of the group <math>G</math>), there exists a Hecke character <math>\chi</math> of <math>K</math> such that
 :<math>L_{E/K}^{\mathrm{Artin}}(\sigma, s) = L_{K}^{\mathrm{Hecke}}(\chi, s)</math>
-where the left hand side is the Artin L-function associated to the extension with character $\sigma$ and the right hand side is the Hecke $L$-function associated with $\chi$
+where the left hand side is the Artin L-function associated to the extension with character <math>\sigma</math> and the right hand side is the Hecke <math>L</math>-function associated with <math>\chi</math>
@@ 88번째 줄: / 88번째 줄: @@
 ==articles==
 * Alexander Polishchuk, A-infinity algebras associated with elliptic curves and Eisenstein-Kronecker series, arXiv:1604.07888 [math.AG], April 26 2016, http://arxiv.org/abs/1604.07888
-* Thorner, Jesse, and Asif Zaman. “Explicit Results on the Distribution of Zeros of Hecke $L$-Functions.” arXiv:1510.08086 [math], October 27, 2015. http://arxiv.org/abs/1510.08086.
+* Thorner, Jesse, and Asif Zaman. “Explicit Results on the Distribution of Zeros of Hecke <math>L</math>-Functions.” arXiv:1510.08086 [math], October 27, 2015. http://arxiv.org/abs/1510.08086.
 * Zaman, Asif. “Explicit Estimates for the Zeros of Hecke L-Functions.” arXiv:1502.05679 [math], February 19, 2015. http://arxiv.org/abs/1502.05679.
 [[분류:L-functions and L-values]]
 [[분류:migrate]]

← 이전 판		2020년 11월 13일 (금) 09:35 판
92번째 줄:		92번째 줄:

	[[분류:L-functions and L-values]]		[[분류:L-functions and L-values]]
		+	[[분류:migrate]]

@@ 87번째 줄: / 87번째 줄: @@
 ==articles==
 * Thorner, Jesse, and Asif Zaman. “Explicit Results on the Distribution of Zeros of Hecke $L$-Functions.” arXiv:1510.08086 [math], October 27, 2015. http://arxiv.org/abs/1510.08086.
 * Zaman, Asif. “Explicit Estimates for the Zeros of Hecke L-Functions.” arXiv:1502.05679 [math], February 19, 2015. http://arxiv.org/abs/1502.05679.
 [[분류:L-functions and L-values]]

← 이전 판		2015년 8월 25일 (화) 08:50 판
66번째 줄:		66번째 줄:
	* {{수학노트\|url=디리클레_L-함수}}		* {{수학노트\|url=디리클레_L-함수}}

		+	==zeta integral==
		+	* [[zeta integral]]

−	~~==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~~
−	$$
−	~~\zeta(f,\chi,s)=\zeta(\widehat{f},\overline{\chi},1-s)~~
−	$$

	==memo==		==memo==

@@ 120번째 줄: / 120번째 줄: @@
 * '''[Leahy2010]''' James-Michael Leahy, [http://www.math.mcgill.ca/darmon/theses/leahy/thesis.pdf An introduction to Tate's Thesis]
 * Herz, Carl, Stephen William Drury, and Maruti Ram Murty. 1997. Harmonic Analysis and Number Theory: Papers in Honour of Carl S. Herz : Proceedings of a Conference on Harmonic Analysis and Number Theory, April 15-19, 1996, McGill University, Montréal, Canada. American Mathematical Soc.
 [[분류:L-functions and L-values]]

← 이전 판		2015년 1월 2일 (금) 09:10 판
120번째 줄:		120번째 줄:
	* '''[Leahy2010]''' James-Michael Leahy, [http://www.math.mcgill.ca/darmon/theses/leahy/thesis.pdf An introduction to Tate's Thesis]		* '''[Leahy2010]''' James-Michael Leahy, [http://www.math.mcgill.ca/darmon/theses/leahy/thesis.pdf An introduction to Tate's Thesis]
	* Herz, Carl, Stephen William Drury, and Maruti Ram Murty. 1997. Harmonic Analysis and Number Theory: Papers in Honour of Carl S. Herz : Proceedings of a Conference on Harmonic Analysis and Number Theory, April 15-19, 1996, McGill University, Montréal, Canada. American Mathematical Soc.		* Herz, Carl, Stephen William Drury, and Maruti Ram Murty. 1997. Harmonic Analysis and Number Theory: Papers in Honour of Carl S. Herz : Proceedings of a Conference on Harmonic Analysis and Number Theory, April 15-19, 1996, McGill University, Montréal, Canada. American Mathematical Soc.
		+	[[분류:L-function]]