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