Bloch-Beilinson conjecture for elliptic curves

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

Beilinson conjectures

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

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Spacy 패턴 목록

[{'LOWER': 'special'}, {'LOWER': 'values'}, {'LOWER': 'of'}, {'LOWER': 'l'}, {'OP': '*'}, {'LEMMA': 'function'}]