Hecke indefinite modular forms - 편집 역사

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example

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example

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imported>Pythagoras0: Pythagoras0 사용자가 Indefinite theta functions 문서를 Hecke indefinite modular forms 문서로 옮겼습니다.

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Pythagoras0 사용자가 Indefinite theta functions 문서를 Hecke indefinite modular forms 문서로 옮겼습니다.

imported>Pythagoras0: /* articles */

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articles

imported>Pythagoras0: /* memo */

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memo

imported>Pythagoras0: /* string functions */

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string functions

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 ==introduction==
-Let $\mathfrak{g} = A_1^{(1)}$. Fix a dominant integral weight $\Lambda$ of $\mathfrak{g}$ of level $m \geq 1$,
+Let <math>\mathfrak{g} = A_1^{(1)}</math>. Fix a dominant integral weight <math>\Lambda</math> of <math>\mathfrak{g}</math> of level <math>m \geq 1</math>,
-and let $\lambda$ be a maximal dominant weight of $L(\Lambda)$.
+and let <math>\lambda</math> be a maximal dominant weight of <math>L(\Lambda)</math>.
-Let $N$ denote the quadratic form defined on $\mathbb{R}^2$ by:
+Let <math>N</math> denote the quadratic form defined on <math>\mathbb{R}^2</math> by:
-$$ N(x,y):= 2 (m+2)x^2  - 2m y^2 \;\;\;\; (x,y \in \mathbb{R})$$
+:<math> N(x,y):= 2 (m+2)x^2  - 2m y^2 \;\;\;\; (x,y \in \mathbb{R})</math>
-and let $(\cdot|\cdot)$ denote the corresponding symmetric bilinear form.
+and let <math>(\cdot|\cdot)</math> denote the corresponding symmetric bilinear form.
-Let $M:=\mathbb{Z}^2$ and let $M^*$ denote the lattice dual to $M$ with respect to this form.
+Let <math>M:=\mathbb{Z}^2</math> and let <math>M^*</math> denote the lattice dual to <math>M</math> with respect to this form.
-Let $O(N)$ denote the group of invertible linear operators on $\mathbb{R}^2$
+Let <math>O(N)</math> denote the group of invertible linear operators on <math>\mathbb{R}^2</math>
-preserving $N$, and $SO_0(N)$ be the connected component of
+preserving <math>N</math>, and <math>SO_0(N)</math> be the connected component of
-$O(N)$ containing the identity. We then have the groups
+<math>O(N)</math> containing the identity. We then have the groups
-$G := \{g \in SO_0(N): g M =M \}$ and $G_0 := \{g \in G: g \text{ fixes } M^*/M \text{ pointwise}\}$.
+<math>G := \{g \in SO_0(N): g M =M \}</math> and <math>G_0 := \{g \in G: g \text{ fixes } M^*/M \text{ pointwise}\}</math>.
-The set $U^+:=\{(x,y) \in \mathbb{R}^2: N(x,y) >0\}$ is preserved under the action of $O(N)$ on $\mathbb{R}^2$.
+The set <math>U^+:=\{(x,y) \in \mathbb{R}^2: N(x,y) >0\}</math> is preserved under the action of <math>O(N)</math> on <math>\mathbb{R}^2</math>.
-We let $A:=\frac{\langle{\Lambda + \rho,\check{\alpha}_1}\rangle}{2(m+2)}$ and
+We let <math>A:=\frac{\langle{\Lambda + \rho,\check{\alpha}_1}\rangle}{2(m+2)}</math> and
-$B:= \frac{\langle{\lambda, \check{\alpha}_1}\rangle}{2m}$ where $\check{\alpha}_1$ is the coroot corresponding to the underlying finite type diagram ($\mathfrak{sl}_2$ in this case), and $\rho$ is the Weyl vector.  Then, $(A,B) \in M^*$, and we set $L:= (A,B) + M$.
+<math>B:= \frac{\langle{\lambda, \check{\alpha}_1}\rangle}{2m}</math> where <math>\check{\alpha}_1</math> is the coroot corresponding to the underlying finite type diagram (<math>\mathfrak{sl}_2</math> in this case), and <math>\rho</math> is the Weyl vector.  Then, <math>(A,B) \in M^*</math>, and we set <math>L:= (A,B) + M</math>.
 The Hecke indefinite modular form is the following sum:
-$$\theta_L(\tau) := \sum_{\substack{(x,y) \in L \cap U^+ \\ (x,y) \text{ mod } G_0}} \mathrm{sign}(x,y) \,   e^{\pi i \tau N(x,y)},$$
+:<math>\theta_L(\tau) := \sum_{\substack{(x,y) \in L \cap U^+ \\ (x,y) \text{ mod } G_0}} \mathrm{sign}(x,y) \,   e^{\pi i \tau N(x,y)},</math>
-where $\mathrm{sign}(x,y) = 1$ for $x \geq 0$ and $-1$ for $x<0$.
+where <math>\mathrm{sign}(x,y) = 1</math> for <math>x \geq 0</math> and <math>-1</math> for <math>x<0</math>.
-This is an absolutely convergent sum for $\tau$ in the upper half plane $\mathbb{H}$, and defines a cusp form of weight 1.
+This is an absolutely convergent sum for <math>\tau</math> in the upper half plane <math>\mathbb{H}</math>, and defines a cusp form of weight 1.
 ===examples===
 * Hecke
-$$
+:<math>
 \prod_{n=1}^{\infty}(1-q^n)^2=\sum_{\substack{m,n=-\infty \\ n\geq 2|m|}}^{\infty}(-1)^{n+m}q^{(n^2-3m^2)/2+(n+m)/2}
-$$
+</math>
 * Kac-Peterson
-$$
+:<math>
 \prod_{n=1}^{\infty}(1-q^n)(1-q^{2n})=\sum_{\substack{m,n=-\infty \\ n\geq 3|m|}}^{\infty}(-1)^{n+m}q^{(n^2-8m^2)/2+n/2}
-$$
+</math>
 ==string functions==
 ;thm (Kac-Peterson)
-Let $\mathfrak{g} = A_1^{(1)}$. Let $\Lambda$ be a dominant integral weight of $\mathfrak{g}$, and $\lambda$ be a maximal dominant weight of $L(\Lambda)$. Then
+Let <math>\mathfrak{g} = A_1^{(1)}</math>. Let <math>\Lambda</math> be a dominant integral weight of <math>\mathfrak{g}</math>, and <math>\lambda</math> be a maximal dominant weight of <math>L(\Lambda)</math>. Then
-$$c^{\Lambda}_{\lambda}(\tau) = \theta_L(\tau) \, \eta(\tau)^{\scriptstyle{s} -3}.$$
+:<math>c^{\Lambda}_{\lambda}(\tau) = \theta_L(\tau) \, \eta(\tau)^{\scriptstyle{s} -3}.</math>
-Here $\theta_L(\tau)$ is a Hecke indefinite modular form and $\eta(\tau)$ is the Dedekind eta function.
+Here <math>\theta_L(\tau)</math> is a Hecke indefinite modular form and <math>\eta(\tau)</math> is the Dedekind eta function.
 ==memo==
@@ 50번째 줄: / 50번째 줄: @@
 ==articles==
 * Westerholt-Raum, Martin. “H-Harmonic Maass-Jacobi Forms of Degree 1: The Analytic Theory of Indefinite Theta Series.” arXiv:1207.5603 [math], July 24, 2012. http://arxiv.org/abs/1207.5603.
-* Sharma, Sachin S., and Sankaran Viswanath. ‘The $t$-Analogs of String Functions for $A_1^{(1)}$ and Hecke Indefinite Modular Forms’. arXiv:1302.6200 [math], 25 February 2013. http://arxiv.org/abs/1302.6200.
+* Sharma, Sachin S., and Sankaran Viswanath. ‘The <math>t</math>-Analogs of String Functions for <math>A_1^{(1)}</math> and Hecke Indefinite Modular Forms’. arXiv:1302.6200 [math], 25 February 2013. http://arxiv.org/abs/1302.6200.
 * Polishchuk, Alexander. ‘A New Look at Hecke’s Indefinite Theta Series’. arXiv:math/0012005, 1 December 2000. http://arxiv.org/abs/math/0012005.
 * Hiramatsu, Toyokazu, Noburo Ishii, and Yoshio Mimura. ‘On Indefinite Modular Forms of Weight One’. Journal of the Mathematical Society of Japan 38, no. 1 (January 1986): 67–83. doi:10.2969/jmsj/03810067.

← 이전 판		2020년 11월 14일 (토) 00:20 판
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	[[분류:mock modular forms]]		[[분류:mock modular forms]]
	[[분류:math]]		[[분류:math]]
		+	[[분류:migrate]]

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 ==articles==
 * Sharma, Sachin S., and Sankaran Viswanath. ‘The $t$-Analogs of String Functions for $A_1^{(1)}$ and Hecke Indefinite Modular Forms’. arXiv:1302.6200 [math], 25 February 2013. http://arxiv.org/abs/1302.6200.
 * Polishchuk, Alexander. ‘A New Look at Hecke’s Indefinite Theta Series’. arXiv:math/0012005, 1 December 2000. http://arxiv.org/abs/math/0012005.

@@ 21번째 줄: / 21번째 줄: @@
 This is an absolutely convergent sum for $\tau$ in the upper half plane $\mathbb{H}$, and defines a cusp form of weight 1.
-===example===
+===examples===
-* we have
+* Hecke
 $$
 \prod_{n=1}^{\infty}(1-q^n)^2=\sum_{\substack{m,n=-\infty \\ n\geq 2|m|}}^{\infty}(-1)^{n+m}q^{(n^2-3m^2)/2+(n+m)/2}
 $$

@@ 24번째 줄: / 24번째 줄: @@
 * we have
 $$
-\prod_{n=1}^{\infty}(1-q^n)^2=\sum_{\substack{m,n=-\infty \\ n\geq 2|m|}}^{\infty}(-1)^{n+m}q^{(n^2-3m^2_/2+(n+m)/2}
+\prod_{n=1}^{\infty}(1-q^n)^2=\sum_{\substack{m,n=-\infty \\ n\geq 2|m|}}^{\infty}(-1)^{n+m}q^{(n^2-3m^2)/2+(n+m)/2}
 $$
 ==string functions==

← 이전 판	2014년 12월 16일 (화) 04:30 판
(차이 없음)

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 * Jimbo, Michio, Tetsuji Miwa, and Masato Okado. ‘Solvable Lattice Models with Broken ZN Symmetry and Hecke’s Indefinite Modular Forms’. Nuclear Physics B 275, no. 3 (24 November 1986): 517–45. doi:[http://dx.doi.org/10.1016/0550-3213(86)90611-5 10.1016/0550-3213(86)90611-5].
 * Jimbo, Michio, and Tetsuji Miwa. ‘A Solvable Lattice Model and Related Rogers-Ramanujan Type Identities’. Physica D: Nonlinear Phenomena 15, no. 3 (April 1985): 335–53. doi:10.1016/S0167-2789(85)80003-8.
 * Kac, V. G., and D. H. Peterson. ‘Affine Lie Algebras and Hecke Modular Forms’. Bulletin (New Series) of the American Mathematical Society 3, no. 3 (November 1980): 1057–61. http://projecteuclid.org/euclid.bams/1183547694
 *  Über einen Zusammenhang zwischen elliptischen Modulfunktionen und indefiniten quadratischen Formen

@@ 28번째 줄: / 28번째 줄: @@
 ==memo==
-* see the appendix of
+* see Appendix of Jimbo, Miwa and Okado, 1986
 ==related items==