"Bloch-Beilinson conjecture for elliptic curves"의 두 판 사이의 차이
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==introduction== | ==introduction== | ||
* In 1986, Spencer Bloch gave an abstract definition of a (regulator) map from higher Chow groups to Deligne-Beilinson cohomology | * In 1986, Spencer Bloch gave an abstract definition of a (regulator) map from higher Chow groups to Deligne-Beilinson cohomology | ||
− | * | + | * <math>E</math>: elliptic curve over <math>\mathbb{Q}</math> |
− | * the value at | + | * the value at <math>s=2</math> of the <math>L</math>-function for <math>E</math> in terms of a regulator map |
− | + | :<math> | |
K_2(E_{\mathbb{C}}) \to \mathbb{C} | K_2(E_{\mathbb{C}}) \to \mathbb{C} | ||
− | + | </math> | |
* When E has complex multiplication a proof of the conjecture has been given by D. Rohrlich | * When E has complex multiplication a proof of the conjecture has been given by D. Rohrlich | ||
;conjecture | ;conjecture | ||
− | Let | + | Let <math>E</math> be an elliptic curve over <math>\mathbb{Q}</math> and <math>\mathcal{E}</math> a neron model of E. Then <math>K_2(\mathcal{E})</math> is of rank 1 and |
− | + | :<math> | |
L'(E,0)\sim_{\mathbb{Q}^{\times}}r(\alpha) | L'(E,0)\sim_{\mathbb{Q}^{\times}}r(\alpha) | ||
− | + | </math> | |
− | for | + | for <math>\alpha\in K_2(\mathcal{E})\backslash K_2(\mathcal{E})_{\mathrm tor}</math> |
− | * there is not a single instance of an elliptic curve | + | * there is not a single instance of an elliptic curve <math>E/\mathbb{Q}</math> for which we know <math>K_2(\mathcal{E})\otimes \mathbb{Q}</math> is one-dimensional (or even finite-dimensional) it is actually quite hard to construct elements in this group |
25번째 줄: | 25번째 줄: | ||
* Weißschuh, Thomas. ‘A Commutative Regulator Map into Deligne-Beilinson Cohomology’. arXiv:1410.4686 [math], 17 October 2014. http://arxiv.org/abs/1410.4686. | * Weißschuh, Thomas. ‘A Commutative Regulator Map into Deligne-Beilinson Cohomology’. arXiv:1410.4686 [math], 17 October 2014. http://arxiv.org/abs/1410.4686. | ||
* Duke, William, and Özlem Imamoḡlu. 2007. “On a Formula of Bloch.” Uniwersytet Im. Adama Mickiewicza W Poznaniu. Wydzia\l\ Matematyki I Informatyki. Functiones et Approximatio Commentarii Mathematici 37 (part 1): 109–117. doi:10.7169/facm/1229618744. | * Duke, William, and Özlem Imamoḡlu. 2007. “On a Formula of Bloch.” Uniwersytet Im. Adama Mickiewicza W Poznaniu. Wydzia\l\ Matematyki I Informatyki. Functiones et Approximatio Commentarii Mathematici 37 (part 1): 109–117. doi:10.7169/facm/1229618744. | ||
− | * Bloch, S., and D. Grayson. 1986. “ | + | * Bloch, S., and D. Grayson. 1986. “<math>K_2</math> and <math>L</math>-Functions of Elliptic Curves: Computer Calculations.” In Applications of Algebraic <math>K</math>-Theory to Algebraic Geometry and Number Theory, Part I, II (Boulder, Colo., 1983), 55:79–88. Contemp. Math. Providence, RI: Amer. Math. Soc. http://www.ams.org/mathscinet-getitem?mr=862631. |
− | * Rohrlich, David E. 1987. “Elliptic Curves and Values of | + | * Rohrlich, David E. 1987. “Elliptic Curves and Values of <math>L</math>-Functions.” In Number Theory (Montreal, Que., 1985), 7:371–387. CMS Conf. Proc. Providence, RI: Amer. Math. Soc. http://www.ams.org/mathscinet-getitem?mr=894330. |
− | * Bloch, Spencer. 1981. “The Dilogarithm and Extensions of Lie Algebras.” In Algebraic | + | * Bloch, Spencer. 1981. “The Dilogarithm and Extensions of Lie Algebras.” In Algebraic <math>K</math>-Theory, Evanston 1980 (Proc. Conf., Northwestern Univ., Evanston, Ill., 1980), 854:1–23. Lecture Notes in Math. Berlin: Springer. http://www.ams.org/mathscinet-getitem?mr=618298. |
− | * Beilinson, A. A. 1980. “Higher Regulators and Values of | + | * 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. |
− | * Bloch, S. 1980. “Algebraic | + | * Bloch, S. 1980. “Algebraic <math>K</math>-Theory and Zeta Functions of Elliptic Curves.” In Proceedings of the International Congress of Mathematicians (Helsinki, 1978), 511–515. Helsinki: Acad. Sci. Fennica. http://www.ams.org/mathscinet-getitem?mr=562648. |
==books== | ==books== |
2020년 11월 16일 (월) 05:28 판
introduction
- In 1986, Spencer Bloch gave an abstract definition of a (regulator) map from higher Chow groups to Deligne-Beilinson cohomology
- \(E\): elliptic curve over \(\mathbb{Q}\)
- the value at \(s=2\) of the \(L\)-function for \(E\) in terms of a regulator map
\[ K_2(E_{\mathbb{C}}) \to \mathbb{C} \]
- When E has complex multiplication a proof of the conjecture has been given by D. Rohrlich
- conjecture
Let \(E\) be an elliptic curve over \(\mathbb{Q}\) and \(\mathcal{E}\) a neron model of E. Then \(K_2(\mathcal{E})\) is of rank 1 and \[ L'(E,0)\sim_{\mathbb{Q}^{\times}}r(\alpha) \] for \(\alpha\in K_2(\mathcal{E})\backslash K_2(\mathcal{E})_{\mathrm tor}\)
- there is not a single instance of an elliptic curve \(E/\mathbb{Q}\) for which we know \(K_2(\mathcal{E})\otimes \mathbb{Q}\) is one-dimensional (or even finite-dimensional) it is actually quite hard to construct elements in this group
articles
- Laterveer, Robert. “A Short Note on the Weak Lefschetz Property for Chow Groups.” arXiv:1507.04485 [math], July 16, 2015. doi:10.1007/s10231-015-0522-y.
- Brunault, François, and Masataka Chida. ‘Regulators for Rankin-Selberg Products of Modular Forms’. arXiv:1503.04626 [math], 16 March 2015. http://arxiv.org/abs/1503.04626.
- Weißschuh, Thomas. ‘A Commutative Regulator Map into Deligne-Beilinson Cohomology’. arXiv:1410.4686 [math], 17 October 2014. http://arxiv.org/abs/1410.4686.
- Duke, William, and Özlem Imamoḡlu. 2007. “On a Formula of Bloch.” Uniwersytet Im. Adama Mickiewicza W Poznaniu. Wydzia\l\ Matematyki I Informatyki. Functiones et Approximatio Commentarii Mathematici 37 (part 1): 109–117. doi:10.7169/facm/1229618744.
- Bloch, S., and D. Grayson. 1986. “\(K_2\) and \(L\)-Functions of Elliptic Curves: Computer Calculations.” In Applications of Algebraic \(K\)-Theory to Algebraic Geometry and Number Theory, Part I, II (Boulder, Colo., 1983), 55:79–88. Contemp. Math. Providence, RI: Amer. Math. Soc. http://www.ams.org/mathscinet-getitem?mr=862631.
- Rohrlich, David E. 1987. “Elliptic Curves and Values of \(L\)-Functions.” In Number Theory (Montreal, Que., 1985), 7:371–387. CMS Conf. Proc. Providence, RI: Amer. Math. Soc. http://www.ams.org/mathscinet-getitem?mr=894330.
- Bloch, Spencer. 1981. “The Dilogarithm and Extensions of Lie Algebras.” In Algebraic \(K\)-Theory, Evanston 1980 (Proc. Conf., Northwestern Univ., Evanston, Ill., 1980), 854:1–23. Lecture Notes in Math. Berlin: Springer. http://www.ams.org/mathscinet-getitem?mr=618298.
- 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.
- Bloch, S. 1980. “Algebraic \(K\)-Theory and Zeta Functions of Elliptic Curves.” In Proceedings of the International Congress of Mathematicians (Helsinki, 1978), 511–515. Helsinki: Acad. Sci. Fennica. http://www.ams.org/mathscinet-getitem?mr=562648.
books
- Higher Regulators, Algebraic K-Theory, and Zeta Functions of Elliptic Curves Bloch, American Mathematical Society