Hostname: page-component-586b7cd67f-r5fsc Total loading time: 0 Render date: 2024-11-25T16:17:24.837Z Has data issue: false hasContentIssue false
  • English
  • Français

Reciprocity Laws on Algebraic Surfaces via Iterated Integrals

Published online by Cambridge University Press:  30 September 2014

Ivan Horozov  and
Matt Kerr
Ivan Horozov
Affiliation:
Washington University in St Louis, Department of Mathematics, Campus Box 1146, One Brookings Drive, St Louis, MO 63130, USA, [email protected]
Matt Kerr
Affiliation:
Washington University in St Louis, Department of Mathematics, Campus Box 1146, One Brookings Drive, St Louis, MO 63130, USA
Get access

Abstract

In this paper we introduce new local symbols, which we call 4-function local symbols. We formulate reciprocity laws for them. These reciprocity laws are proven using a new method - multidimensional iterated integrals. Besides providing reciprocity laws for the new 4-function local symbols, the same method works for proving reciprocity laws for the Parshin symbol. Both the new 4-function local symbols and the Parshin symbol can be expressed as a finite product of newly defined bi-local symbols, each of which satisfies a reciprocity law. The K-theoretic variant of the first 4-function local symbol is defined in the Appendix. It differs by a sign from the one defined via iterated integrals. Both the sign and the K-theoretic variant of the 4-function local symbol satisfy reciprocity laws, whose proof is based on Milnor K-theory (see the Appendix). The relation of the 4-function local symbols to the double free loop space of the surface is given by iterated integrals over membranes.

Keywords

reciprocity lawscomplex algebraic surfacesiterated integrals14C3032J2555P35
Type
Research Article
Information
Journal of K-Theory , Volume 14 , Issue 2 , October 2014 , pp. 273 - 312
Copyright
Copyright © ISOPP 2014 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

BrMcL.Brylinski, J.-L.; McLaughlin, D. A.: The geometry of two-dimensional symbols. K-Theory 10(3) (1996), 215–237.Google Scholar
Chi.Chen, K.-T.: Iterated path integrals, Bull. AMS 83(1977), 831–879.Google Scholar
Ch2.Chen, K.-T.: Iterated integrals of differential forms and loop space homology, Ann. Math. 97 (1973), 217–246.Google Scholar
Co.Contou-Carrère, C.: Jacobienne locale, groupe de bivecteurs de Witt universel et symbole modéré, C. R. Acad. Sci. Paris Sér I 318 (1994), 743–746.Google Scholar
D1.Deligne, P.: Le symbole modéré. Inst. Hautes Études Sci. Publ. Math. 73 (1991), 147–181.Google Scholar
D2.Deligne, P.: email to the author.Google Scholar
FV.Fesenko, I., Vostokov, S.On torsion in higher Milnor functors for multidimensional local fields, Amer. Math. Soc Transl. 2, 154 (1992), 25–35.Google Scholar
GH.Griffiths, P., Harris, J.: Principles of algebraic geometry, Pure and Applied Mathematics. Wiley-Interscience [John Wiley & Sons], New York, 1978. xii + 813 pp.Google Scholar
Hi.Horozov, I.: Non-abelian reciprocity laws on a Riemann surface. Int. Math. Res. Not. IMRN 11 (2011), 2469–2495.Google Scholar
H2.Horozov, I.: A refinement of the Parshin symbol for surfaces, arXiv:1002.2698.Google Scholar
H3.Horozov, I.: Non-commutative Two Dimensional Modular Symbol, arXiv:math/0611955.Google Scholar
H4.Horozov, I.: Multiple Dedekind Zeta Functions, arXiv:1101. 1594 [math. NT], to appear in Journal für die reine und angew. Math.Google Scholar
H5.Horozov, I.: Non-commutative Hilbert Modular Symbols, arXiv:1308. 4991 [math. NT], 33 pages, pending in Algebra and Number Theory.Google Scholar
H6.Horozov, I.Luo, Zh.: Two dimensional Contou-Carrère symbol via iterated integrals, in preparation.Google Scholar
Ka.Kato, K.: Milnor K-theory and the Chow group of zero cycles, Contemp. Math. 55, A. M. S. (1986), 241–253.Google Scholar
Ke1.Kerr, M.: An elementary proof of Suslin reciprocity, Canad. Math. Bull. 48(2) (2005), 221–236.CrossRefGoogle Scholar
Ke2.Kerr, M.: Geometric construction of regulator currents with applications to algebraic cycles, Princeton University Ph. D. Thesis, 2003.Google Scholar
Kh.Khovanskii, Askold: Logarithmic functional and reciprocity laws. Toric topology, 221-229, Contemp. Math. 460, Amer. Math. Soc., Providence, RI, 2008.Google Scholar
K1.Kleiman, S.: Geometry on Grassmannians and applications to splitting bundles and smoothing cycles. Publ. Math., Inst. Hautes Étud. Sci. 36 (1969), 281–297.CrossRefGoogle Scholar
M1.Manin, Yu. I.: Iterated integrals of modular forms and non-commutative modular symbols, Algebraic Geometry and Number Theory, in Honor of Vladimir Drinfeld's 50th Birthday, Ginzburg, V. (ed. ), Progress in Math. 256, Birkhäuser Boston, Boston, 2006, pp. 565–597; preprint AG/0502576. math. NT/0502576, 37 pages.Google Scholar
M2.Manin, Yu. I.: email to the first author.Google Scholar
Mi.Milnor, J.: Introduction to Algebraic K-Theory, Annals of Mathematics Studies, Princeton University Press, 1971.Google Scholar
O.Osipov, D.: email to the author.Google Scholar
OZh.Osipov, Denis; Zhu, Xinwen: A categorical proof of the Parshin reciprocity laws on algebraic surfaces. Algebra and Number Theory 5(3) (2011), 289–337.Google Scholar
P1.Parshin, A. N.: Local class field theory, Trudy Mat. Inst. Steklov 165, 1984.Google Scholar
P2.Parshin, A. N.: Galois cohomology and Brauer group of local fields, Trudy Mat. Inst. Steklov 183, 1984.Google Scholar
PR1.Pablo Romo, F.: Algebraic construction of the tame symbol and the Parshin symbol on a surface, Algebra 274(1) (2004), 335–346.Google Scholar
PR2.Pablo Romo, F.: A General Reciprocity Law for Symbols on Arbitrary Vector Spaces, arXiv:1305. 7066 [math. NT].Google Scholar
Ro.Rost, M.: Chow groups with coefficients, Doc. Math. 1(16) (1996), 319–393.CrossRefGoogle Scholar
Th.Thomas, R. P.: Nodes and the Hodge conjecture. J. Algebraic Geometry 14 (2005), 177–185.Google Scholar
W.Weil, A.: Sur les fonctions algébriques á corps de constantes fini. C. R. Acad. Sci. Paris 210 (1940), 592–594.Google Scholar