Hostname: page-component-cd9895bd7-dk4vv Total loading time: 0 Render date: 2024-12-25T04:34:17.200Z Has data issue: false hasContentIssue false
  • English
  • Français

K1 of products of Drinfeld modular curves and special values of L-functions

Part of: Arithmetic algebraic geometry Discontinuous groups and automorphic forms

Published online by Cambridge University Press:  08 June 2010

Ramesh Sreekantan
Ramesh Sreekantan*
Affiliation:
Indian Statistical Institute, 8th Mile, Mysore Road, Jnana Bharathi, Bangalore 560 059, India (email: [email protected])
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Beilinson [Higher regulators and values of L-functions, Itogi Nauki i Tekhniki Seriya Sovremennye Problemy Matematiki Noveishie Dostizheniya (Current problems in mathematics), vol. 24 (Vserossiisky Institut Nauchnoi i Tekhnicheskoi Informatsii, Moscow, 1984), 181–238] obtained a formula relating the special value of the L-function of H2 of a product of modular curves to the regulator of an element of a motivic cohomology group, thus providing evidence for his general conjectures on special values of L-functions. In this paper we prove a similar formula for the L-function of the product of two Drinfeld modular curves, providing evidence for an analogous conjecture in the case of function fields.

Keywords

Drinfeld modular curvesmotivic cohomologyspecial values of L-functions

MSC classification

Secondary: 11F52: Modular forms associated to Drinfel'd modules 11G40: $L$-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture
Type
Research Article
Information
Compositio Mathematica , Volume 146 , Issue 4 , July 2010 , pp. 886 - 918
Copyright
Copyright © Foundation Compositio Mathematica 2010

References

[1]Baba, S. and Sreekantan, R., An analogue of circular units for products of elliptic curves, Proc. Edinb. Math. Soc. (2) 47 (2004), 35–51.CrossRefGoogle Scholar
[2]Beĭlinson, A. A., Higher regulators and values of L-functions, Itogi Nauki i Tekhniki Seriya Sovremennye Problemy Matematiki Noveishie Dostizheniya (Current problems in mathematics), vol. 24 (Vserossiisky Institut Nauchnoi i Tekhnicheskoi Informatsii, Moscow, 1984), 181–238.Google Scholar
[3]Bloch, S., Algebraic cycles and higher K-theory, Adv. Math. 61 (1986), 267–304.CrossRefGoogle Scholar
[4]Bloch, S., Gillet, H. and Soulé, C., Non-Archimedean Arakelov theory, J. Algebraic Geom. 4 (1995), 427–485.Google Scholar
[5]Consani, C., Double complexes and Euler L-factors, Compositio Math. 111 (1998), 323–358.CrossRefGoogle Scholar
[6]Consani, C., The local monodromy as a generalized algebraic correspondence, Doc. Math. 4 (1999), 65–108. With an appendix by Spencer Bloch (electronic).CrossRefGoogle Scholar
[7]Deligne, P., Les constantes des équations fonctionnelles des L-functions, in Modular functions of one variable, II (Proc. Int. Summer School, University of Antwerp) Antwerp, 17 July–3 August 1972, Lecture Notes in Mathematics, vol. 349 (Springer, Berlin, 1973), 501–597.Google Scholar
[8]Gekeler, E.-U., Improper Eisenstein series on Bruhat–Tits trees, Manuscripta Math. 86 (1995), 367–391.CrossRefGoogle Scholar
[9]Gekeler, E.-U., On the Drinfeld discriminant function, Compositio Math. 106 (1997), 181–202.CrossRefGoogle Scholar
[10]Gekeler, E.-U. and Reversat, M., Jacobians of Drinfeld modular curves, J. Reine Angew. Math. 476 (1996), 27–93.Google Scholar
[11]Ogg, A. P., On a convolution of L-series, Invent. Math. 7 (1969), 297–312.CrossRefGoogle Scholar
[12]Papikian, M., On the degree of modular parametrizations over function fields, J. Number Theory 97 (2002), 317–349.CrossRefGoogle Scholar
[13]Ramakrishnan, D., Regulators, algebraic cycles, and values of L-functions, in Algebraic K- theory and algebraic number theory, Honolulu, HI, 12–16 January 1987, Contemporary Mathematics, vol. 83 (American Mathematical Society, Providence, RI, 1989), 183–310.Google Scholar
[14]Rapoport, M., Schappacher, N. and Schneider, P. (eds), Beilinson’s conjectures on special values of L-functions, Perspectives in Mathematics, vol. 4 (Academic Press, Boston, MA, 1988).Google Scholar
[15]Sreekantan, R., A non-Archimedean analogue of the Hodge-𝒟-conjecture for products of elliptic curves, J. Algebraic Geom. 17 (2008), 781–798.CrossRefGoogle Scholar
[16]Teitelbaum, J. T., Modular symbols for Fq(T), Duke Math. J. 68 (1992), 271–295.CrossRefGoogle Scholar