Beilinson conjectures - 편집 역사

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Pythagoras0: /* 메타데이터 */ 새 문단

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메타데이터: 새 문단

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imported>Pythagoras0: /* related items */

2015-03-30T05:48:57Z

imported>Pythagoras0: /* expositions */

2015-01-27T07:12:43Z

expositions

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2015-01-27T05:52:49Z

imported>Pythagoras0: /* review of Dirichler class number formula */

2015-01-25T14:13:25Z

review of Dirichler class number formula

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 [[분류:migrate]]
-== 메타데이터 ==
+==메타데이터==
 ===위키데이터===
 * ID :  [https://www.wikidata.org/wiki/Q7574878 Q7574878]

← 이전 판		2020년 12월 28일 (월) 07:41 판
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	[[분류:L-functions and L-values]]		[[분류:L-functions and L-values]]
	[[분류:migrate]]		[[분류:migrate]]
		+
		+	== 메타데이터 ==
		+
		+	===위키데이터===
		+	* ID : [https://www.wikidata.org/wiki/Q7574878 Q7574878]

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 # Bloch's conjecture about K2 of elliptic curves
 * the Beĭlinson conjectures describe the leading coefficients of L-series of varieties over number fields up to rational factors in terms of generalized regulators
-** the very general setting being for L-functions $L(s)$ associated to Chow motives over number fields
+** the very general setting being for L-functions <math>L(s)</math> associated to Chow motives over number fields
 * Bloch-Beilinson conjecture predicts that ranks of Chow groups of homologically trivial cycles should be related to orders of vanishing of L-functions.
@@ 13번째 줄: / 13번째 줄: @@
 ==motivation==
 * [[L-함수의 값 구하기 입문]]
-* Leibniz formula for $\pi$
+* Leibniz formula for <math>\pi</math>
 :<math>1 \,-\, \frac{1}{3} \,+\, \frac{1}{5} \,-\, \frac{1}{7} \,+\, \frac{1}{9} \,-\, \cdots \;=\; \frac{\pi}{4}</math>
-* the left-hand side is also $L(1)$ where $L(s)$ is the Dirichlet L-function for the Gaussian field.
+* the left-hand side is also <math>L(1)</math> where <math>L(s)</math> is the Dirichlet L-function for the Gaussian field.
 * This formula is a special case of the analytic class number formula, and in those terms reads that the Gaussian field has class number 1, and also contains four roots of unity, so accounting for the factor 1/4.
-* Q : how to replace $\pi$ in the Leibniz formula by some other "transcendental" number
+* Q : how to replace <math>\pi</math> in the Leibniz formula by some other "transcendental" number
 * Beilinson : abstract from the regulator of a number field to some "higher regulator" (the Beilinson regulator), a determinant constructed on a real vector space that comes from algebraic K-theory.
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 ==review of Dirichlet class number formula==
 * {{수학노트|url=이차 수체에 대한 디리클레 유수 (class number) 공식}}
-* let $F$ be a real quadratic field
+* let <math>F</math> be a real quadratic field
 * Dirichlet's class number formula implies
-$$
+:<math>
 \zeta'_F(0)\sim_{\mathbb{Q}^{\times}}\log(|\epsilon|),\, \epsilon \in \mathcal{O}_{F}^{\times},\,\epsilon\neq \pm 1 \label{cls}
-$$
+</math>
 * there are four ingredients
-** an arithmetic/geometric objects ($\mathcal{O}$)
+** an arithmetic/geometric objects (<math>\mathcal{O}</math>)
-** an associated finitely generated abelian group of rank one ($K(\mathcal{O})$)
+** an associated finitely generated abelian group of rank one (<math>K(\mathcal{O})</math>)
-** as associated $L$-function $L(\mathcal{O},s)$ with a simple zero at $s=0$ (the zeta function $\zeta_{F}(s)$
+** as associated <math>L</math>-function <math>L(\mathcal{O},s)</math> with a simple zero at <math>s=0</math> (the zeta function <math>\zeta_{F}(s)</math>
-** a non-zero homomorphism $r:K(\mathcal{O})\to \mathbb{R}$
+** a non-zero homomorphism <math>r:K(\mathcal{O})\to \mathbb{R}</math>
 * with this setup \ref{cls} becomes
-$$
+:<math>
 L'(\mathcal{O},0)\sim_{\mathbb{Q}^{\times}} r(\alpha),\, \alpha \in K(\mathcal{O})\backslash K(\mathcal{O})_{\operatorname{tor}}
-$$
+</math>
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 * seminal 1979 paper
 ;def
-The critical values of a (motivic) $L$-function $L(s)$ are the integers arguments of $s$ at which neither $\gamma(s)$ nor $\gamma(k-s)$ has a pole, where $\gamma(s)$ and $k$ are defined as $L^{*}(s)=L(s)\gamma(s)$ satisfying
+The critical values of a (motivic) <math>L</math>-function <math>L(s)</math> are the integers arguments of <math>s</math> at which neither <math>\gamma(s)</math> nor <math>\gamma(k-s)</math> has a pole, where <math>\gamma(s)</math> and <math>k</math> are defined as <math>L^{*}(s)=L(s)\gamma(s)</math> satisfying
-$$
+:<math>
 L^{*}(s)=\pm L^{*}(k-s)
-$$
+</math>
 * for example, for the Riemann zeta function the critical values are the positive even integers and the negative even integers
 ;conjecture (Deligne, 1979)
-The value of $L(s)$ at any critical value is a non-zero algebraic multiple of the determinant of a certain matrix whose entries are periods
+The value of <math>L(s)</math> at any critical value is a non-zero algebraic multiple of the determinant of a certain matrix whose entries are periods
 ==Beilinson's conjecture==
 * huge generalization of Deligne's conjecture
-* all values of motivic $L$-functions in terms of periods on the variety defining the $L$-function and of a regulator
+* all values of motivic <math>L</math>-functions in terms of periods on the variety defining the <math>L</math>-function and of a regulator
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 ;conjecture (Deligne-Beilinson-Scholl)
-Let $L(s)$ be a motivic $L$-function, $m$ an arbitrary integer, and $r$ the order of vanishing of $L(s)$ at $s=m$. Then $L^{(r)}(m)\in \hat{\mathcal{P}}$ where $\hat{\mathcal{P}}$ denotes the extended period ring.
+Let <math>L(s)</math> be a motivic <math>L</math>-function, <math>m</math> an arbitrary integer, and <math>r</math> the order of vanishing of <math>L(s)</math> at <math>s=m</math>. Then <math>L^{(r)}(m)\in \hat{\mathcal{P}}</math> where <math>\hat{\mathcal{P}}</math> denotes the extended period ring.
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 * [http://perso.ens-lyon.fr/frederic.deglise/Bloch-Kato/programme.pdf Summer school of special values of L-functions]
 * Nekovár, Jan. "Beilinson’s conjectures." U. Jannsen, SL Kleiman, J.–P. Serre,“Motives”, Proceedings of the Research Conference on Motives held July. 1994. http://www.math.jussieu.fr/~nekovar/pu/mot.pdf
-* Scholl, A. J. 1992. “Modular Forms and Algebraic $K$-Theory.” Astérisque (209): 12, 85–97.
+* Scholl, A. J. 1992. “Modular Forms and Algebraic <math>K</math>-Theory.” Astérisque (209): 12, 85–97.
-* Deninger, Christopher, and Anthony J. Scholl. 1991. “The Beilinson Conjectures.” In $L$-Functions and Arithmetic (Durham, 1989), 153:173–209. London Math. Soc. Lecture Note Ser. Cambridge: Cambridge Univ. Press. http://www.ams.org/mathscinet-getitem?mr=1110393.
+* Deninger, Christopher, and Anthony J. Scholl. 1991. “The Beilinson Conjectures.” In <math>L</math>-Functions and Arithmetic (Durham, 1989), 153:173–209. London Math. Soc. Lecture Note Ser. Cambridge: Cambridge Univ. Press. http://www.ams.org/mathscinet-getitem?mr=1110393.
 * [http://wwwmath.uni-muenster.de/u/pschnei/publ/beilinson-volume/Schneider.pdf Introduction to the Beilinson Conjectures], Peter Schneider
@@ 101번째 줄: / 101번째 줄: @@
 * Otsubo, Noriyuki. “On Special Values of Jacobi-Sum Hecke L-Functions.” arXiv:1404.7476 [math], April 29, 2014. http://arxiv.org/abs/1404.7476.
 * Brunault, François. 2006. “Version Explicite Du Théorème de Beilinson Pour La Courbe Modulaire.” Comptes Rendus Mathematique 343 (8) (October 15): 505–510. doi:10.1016/j.crma.2006.09.014.
-* Beilinson, A. A. 1987. “Height Pairing between Algebraic Cycles.” In $K$-Theory, Arithmetic and Geometry (Moscow, 1984–1986), 1289:1–25. Lecture Notes in Math. Berlin: Springer. http://www.ams.org/mathscinet-getitem?mr=923131.
+* Beilinson, A. A. 1987. “Height Pairing between Algebraic Cycles.” In <math>K</math>-Theory, Arithmetic and Geometry (Moscow, 1984–1986), 1289:1–25. Lecture Notes in Math. Berlin: Springer. http://www.ams.org/mathscinet-getitem?mr=923131.
-* Beilinson, A. A. 1984. “Higher Regulators and Values of $L$-Functions.” In Current Problems in Mathematics, Vol. 24, 181–238. Itogi Nauki I Tekhniki. Moscow: Akad. Nauk SSSR Vsesoyuz. Inst. Nauchn. i Tekhn. Inform. http://www.ams.org/mathscinet-getitem?mr=760999. http://dx.doi.org/10.1007/BF02105861
+* Beilinson, A. A. 1984. “Higher Regulators and Values of <math>L</math>-Functions.” In Current Problems in Mathematics, Vol. 24, 181–238. Itogi Nauki I Tekhniki. Moscow: Akad. Nauk SSSR Vsesoyuz. Inst. Nauchn. i Tekhn. Inform. http://www.ams.org/mathscinet-getitem?mr=760999. http://dx.doi.org/10.1007/BF02105861
-* Beilinson, A. A. 1980. “Higher Regulators and Values of $L$-Functions of Curves.” Akademiya Nauk SSSR. Funktsional\cprime Ny\uı\ Analiz I Ego Prilozheniya 14 (2): 46–47.
+* Beilinson, A. A. 1980. “Higher Regulators and Values of <math>L</math>-Functions of Curves.” Akademiya Nauk SSSR. Funktsional\cprime Ny\uı\ Analiz I Ego Prilozheniya 14 (2): 46–47.
 * Deligne, [http://www.math.tifr.res.in/~eghate/Deligne.pdf Values of L-Functions and Periods of Integrals]
 ** translation of P. Deligne’s Valeurs de Fonctions L et periodes d'integrales, in the Proceedings of Symposia in Pure Mathematics 33, (1979), Part 2, 313-346

← 이전 판		2020년 11월 13일 (금) 13:40 판
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	[[분류:K-theory]]		[[분류:K-theory]]
	[[분류:L-functions and L-values]]		[[분류:L-functions and L-values]]
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 * [[Bloch-Beilinson conjecture for elliptic curves]]
 * [[Hypergeometric motives]]
 ==question and answers(Math Overflow)==

← 이전 판	2015년 1월 27일 (화) 06:06 판
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	+	==Beilinson's conjecture==
	+	* huge generalization of Deligne's conjecture
	+	* all values of motivic $L$-functions in terms of periods on the variety defining the $L$-function and of a regulator

	+
	+	==Scholl==
	+	* observed that this regulator can itself be expressed in terms of periods
	+	* this led to a reformuation of Beilinson's conjecture
	+
	+
	+	;conjecture (Deligne-Beilinson-Scholl)
	+	Let $L(s)$ be a motivic $L$-function, $m$ an arbitrary integer, and $r$ the order of vanishing of $L(s)$ at $s=m$. Then $L^{(r)}(m)\in \hat{\mathcal{P}}$ where $\hat{\mathcal{P}}$ denotes the extended period ring.

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 * Beilinson, A. A. 1984. “Higher Regulators and Values of $L$-Functions.” In Current Problems in Mathematics, Vol. 24, 181–238. Itogi Nauki I Tekhniki. Moscow: Akad. Nauk SSSR Vsesoyuz. Inst. Nauchn. i Tekhn. Inform. http://www.ams.org/mathscinet-getitem?mr=760999. http://dx.doi.org/10.1007/BF02105861
 * Beilinson, A. A. 1980. “Higher Regulators and Values of $L$-Functions of Curves.” Akademiya Nauk SSSR. Funktsional\cprime Ny\uı\ Analiz I Ego Prilozheniya 14 (2): 46–47.
 [[분류:K-theory]]
 [[분류:L-functions and L-values]]