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.

motivation

• L-함수의 값 구하기 입문
• Leibniz formula for $\pi$

\[1 \,-\, \frac{1}{3} \,+\, \frac{1}{5} \,-\, \frac{1}{7} \,+\, \frac{1}{9} \,-\, \cdots \;=\; \frac{\pi}{4}\]

• the left-hand side is also $L(1)$ where $L(s)$ is the Dirichlet L-function for the Gaussian field.
• This formula is a special case of the analytic class number formula, and in those terms reads that the Gaussian field has class number 1, and also contains four roots of unity, so accounting for the factor 1/4.
• Q : how to replace $\pi$ in the Leibniz formula by some other "transcendental" number
• Beilinson : abstract from the regulator of a number field to some "higher regulator" (the Beilinson regulator), a determinant constructed on a real vector space that comes from algebraic K-theory.

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.