Beilinson conjectures

introduction

• generalizations of

1. the Lichtenbaum conjectures for K-groups of number rings
2. the Hodge conjecture
3. the Tate conjecture about algebraic cycles
4. the Birch and Swinnerton-Dyer conjecture about elliptic curves
5. Bloch's conjecture about K2 of elliptic curves

• the Beĭlinson conjectures describe the leading coefficients of L-series of varieties over number fields up to rational factors in terms of generalized regulators
  • the very general setting being for L-functions $L(s)$ associated to Chow motives over number fields
• Bloch-Beilinson conjecture predicts that ranks of Chow groups of homologically trivial cycles should be related to orders of vanishing of L-functions.

related items

• Bloch–Kato conjectures for special values of L-functions
• Borel's regulator
• Bloch's regulator
• Hypergeometric motives

question and answers(Math Overflow)

• http://mathoverflow.net/questions/2132/what-is-the-beilinson-regulator

expositions

• Nekovár, Jan. "Beilinson’s conjectures." U. Jannsen, SL Kleiman, J.–P. Serre,“Motives”, Proceedings of the Research Conference on Motives held July. 1994. http://www.math.jussieu.fr/~nekovar/pu/mot.pdf
• Scholl, A. J. 1992. “Modular Forms and Algebraic $K$-Theory.” Astérisque (209): 12, 85–97.
• Deninger, Christopher, and Anthony J. Scholl. 1991. “The Beilinson Conjectures.” In $L$-Functions and Arithmetic (Durham, 1989), 153:173–209. London Math. Soc. Lecture Note Ser. Cambridge: Cambridge Univ. Press. http://www.ams.org/mathscinet-getitem?mr=1110393.
• Introduction to the Beilinson Conjectures, Peter Schneider

articles

• Lemma, Francesco. “On Higher Regulators of Siegel Threefolds II: The Connection to the Special Value.” arXiv:1409.8391 [math], September 30, 2014. http://arxiv.org/abs/1409.8391.
• Miyazaki, Hiroyasu. “Special Values of Zeta Functions of Varieties over Finite Fields via Higher Chow Groups.” arXiv:1406.1390 [math], June 5, 2014. http://arxiv.org/abs/1406.1390.
• Otsubo, Noriyuki. “On Special Values of Jacobi-Sum Hecke L-Functions.” arXiv:1404.7476 [math], April 29, 2014. http://arxiv.org/abs/1404.7476.
• Brunault, François. 2006. “Version Explicite Du Théorème de Beilinson Pour La Courbe Modulaire.” Comptes Rendus Mathematique 343 (8) (October 15): 505–510. doi:10.1016/j.crma.2006.09.014.
• Beilinson, A. A. 1987. “Height Pairing between Algebraic Cycles.” In $K$-Theory, Arithmetic and Geometry (Moscow, 1984–1986), 1289:1–25. Lecture Notes in Math. Berlin: Springer. http://www.ams.org/mathscinet-getitem?mr=923131.
• Beilinson, A. A. 1984. “Higher Regulators and Values of $L$-Functions.” In Current Problems in Mathematics, Vol. 24, 181–238. Itogi Nauki I Tekhniki. Moscow: Akad. Nauk SSSR Vsesoyuz. Inst. Nauchn. i Tekhn. Inform. http://www.ams.org/mathscinet-getitem?mr=760999. http://dx.doi.org/10.1007/BF02105861
• 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.