"Kohnen-Waldspurger formula"의 두 판 사이의 차이
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imported>Pythagoras0 |
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* chapter 15 of [[Unearthing the visions of a master: harmonic Maass forms and number theory]] | * chapter 15 of [[Unearthing the visions of a master: harmonic Maass forms and number theory]] | ||
* study central values and derivatives of weight 2 modular L-functions | * study central values and derivatives of weight 2 modular L-functions | ||
| − | * | + | * let $g(z)$ be a Kohnen newform |
| − | + | * There is a unique newform, say $f(z)\in S^{\text{new}}_{2k}(N)$, associated to $g(z)$ under Shimura's correspondence. | |
| − | * | + | * The coefficients of $g(z)$ determine the central critical values of many of the quadratic twists $L(f, \chi_D, s)$ |
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| 11번째 줄: | 10번째 줄: | ||
* There is a modular form $g(z)=\sum b_{E}(n)q^n$ such that if $\epsilon()=1$, | * There is a modular form $g(z)=\sum b_{E}(n)q^n$ such that if $\epsilon()=1$, | ||
:<math>L(E(\Delta),1)=\alpha(\cdots)b_{E}(\Delta)^{2}</math> | :<math>L(E(\Delta),1)=\alpha(\cdots)b_{E}(\Delta)^{2}</math> | ||
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| + | ==history== | ||
| + | * In the 1980s, Waldspurger [205], and Kohnen and Zagier [135, 136, 137] established that certain half-integral weight modular forms serve as generating functions of a new type. | ||
| + | * Using the Shimura correspondence [192], they proved that certain coefficients of half-integral weight cusp forms essentially are square-roots of central values of quadratic twists of modular L-functions. | ||
| + | * Ono and Bruinier [67] have generalized this theorem of Waldspurger and Kohnen to prove that the Fourier coefficients of weight 1/2 harmonic Maass forms encode the vanishing and nonvanishing of both the central values and derivatives of quadratic twists of weight 2 modular L-functions. | ||
| + | ** [[Mock modular periods and L-functions]] | ||
| 18번째 줄: | 24번째 줄: | ||
==articles== | ==articles== | ||
| − | * [67] [http://arxiv.org/abs/0710.0283 | + | * [67] Bruinier, Jan H., and Ken Ono. 2007. “Heegner Divisors, $L$-Functions and Harmonic Weak Maass Forms.” arXiv:0710.0283 [math] (October 1). http://arxiv.org/abs/0710.0283., Annals of Mathematics |
* [135] W. Kohnen, Newforms of half-integral weight, J. Reine Angew. Math. 333 (1982) 32-72. | * [135] W. Kohnen, Newforms of half-integral weight, J. Reine Angew. Math. 333 (1982) 32-72. | ||
* [136] W. Kohnen, Fourier coefficients of modular forms of half-integral weight, Math. Ann. 271 (1985), 237-268. http://www.springerlink.com/content/p52527460724p36m/ | * [136] W. Kohnen, Fourier coefficients of modular forms of half-integral weight, Math. Ann. 271 (1985), 237-268. http://www.springerlink.com/content/p52527460724p36m/ | ||
* [137] W. Kohnen and D. Zagier, Values of L-series of modular forms at the center of the critical strip, Invent. Math. 64 (1981), no. 2, pages 175–198. | * [137] W. Kohnen and D. Zagier, Values of L-series of modular forms at the center of the critical strip, Invent. Math. 64 (1981), no. 2, pages 175–198. | ||
* [205] J.-L. Waldspurger, Sur les coefficients de Fourier des formes modulaires de poids demi-entier, J. Math. Pures Appl. (9) 60 (1981), no. 4, pages 375–484. | * [205] J.-L. Waldspurger, Sur les coefficients de Fourier des formes modulaires de poids demi-entier, J. Math. Pures Appl. (9) 60 (1981), no. 4, pages 375–484. | ||
2013년 12월 30일 (월) 00:57 판
introduction
- chapter 15 of Unearthing the visions of a master: harmonic Maass forms and number theory
- study central values and derivatives of weight 2 modular L-functions
- let $g(z)$ be a Kohnen newform
- There is a unique newform, say $f(z)\in S^{\text{new}}_{2k}(N)$, associated to $g(z)$ under Shimura's correspondence.
- The coefficients of $g(z)$ determine the central critical values of many of the quadratic twists $L(f, \chi_D, s)$
formula
- There is a modular form $g(z)=\sum b_{E}(n)q^n$ such that if $\epsilon()=1$,
\[L(E(\Delta),1)=\alpha(\cdots)b_{E}(\Delta)^{2}\]
history
- In the 1980s, Waldspurger [205], and Kohnen and Zagier [135, 136, 137] established that certain half-integral weight modular forms serve as generating functions of a new type.
- Using the Shimura correspondence [192], they proved that certain coefficients of half-integral weight cusp forms essentially are square-roots of central values of quadratic twists of modular L-functions.
- Ono and Bruinier [67] have generalized this theorem of Waldspurger and Kohnen to prove that the Fourier coefficients of weight 1/2 harmonic Maass forms encode the vanishing and nonvanishing of both the central values and derivatives of quadratic twists of weight 2 modular L-functions.
articles
- [67] Bruinier, Jan H., and Ken Ono. 2007. “Heegner Divisors, $L$-Functions and Harmonic Weak Maass Forms.” arXiv:0710.0283 [math] (October 1). http://arxiv.org/abs/0710.0283., Annals of Mathematics
- [135] W. Kohnen, Newforms of half-integral weight, J. Reine Angew. Math. 333 (1982) 32-72.
- [136] W. Kohnen, Fourier coefficients of modular forms of half-integral weight, Math. Ann. 271 (1985), 237-268. http://www.springerlink.com/content/p52527460724p36m/
- [137] W. Kohnen and D. Zagier, Values of L-series of modular forms at the center of the critical strip, Invent. Math. 64 (1981), no. 2, pages 175–198.
- [205] J.-L. Waldspurger, Sur les coefficients de Fourier des formes modulaires de poids demi-entier, J. Math. Pures Appl. (9) 60 (1981), no. 4, pages 375–484.