"Bloch-Beilinson conjecture for elliptic curves"의 두 판 사이의 차이

수학노트
둘러보기로 가기 검색하러 가기
imported>Pythagoras0
1번째 줄: 1번째 줄:
 
==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
−
* $E$: elliptic curve over $\mathbb{Q}$
+
* <math>E</math>: elliptic curve over <math>\mathbb{Q}</math>
−
* the value at $s=2$ of the $L$-function for $E$ in terms of a regulator map
+
* 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 $E$ 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  
+
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 $\alpha\in K_2(\mathcal{E})\backslash K_2(\mathcal{E})_{\mathrm tor}$
+
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 $E/\mathbb{Q}$ 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
+
* 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. “$K_2$ 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.
+
* 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 $L$-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.
+
* 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 $K$-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.
+
* 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 $L$-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.
−
* 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.
+
* 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) 04:28 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


related items


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

원본 주소 "https://wiki.mathnt.net/index.php?title=Bloch-Beilinson_conjecture_for_elliptic_curves&oldid=44928"