Abstract
We give explicit equations for the simplest towers of Drinfeld modular curves over any finite field, and observe that they coincide with the asymptotically optimal towers of curves constructed by Garcia and Stichtenoth.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
S. G. Drinfeld and S. G. Vl¨¢dutt: The number of points of an algebraic curve.Functional Anal. Appl.17 (1983), #1, 53–54 (translated from the Russian paper inFunktsional. Anal. i Prilozhen).
N. D. Elkies: Linearized algebra and finite groups of Lie type, I: Linear and symplectic groups. Pages 77–108 inApplications of Curves over Finite Fields(1997 AMS-IMSSIAM Joint Summer Research Conference, July 1997, Washington, Seattle; M. Fried, ed.; Providence: AMS, 1999) =Contemp. Math.245.
N. D. Elkies: Explicit modular towers. Pages 23–32 in Proceedings of the Thirty-Fifth Annual Allerton Conference on Communication, Control and Computing (1997, T. Basar, A. Vardy, eds.), Univ. of Illinois at Urbana-Champaign 1998.
A. Garcia and H. Stichtenoth: A tower of Artin-Schreier extensions of function fields attaining the Drinfeld-Vl¨¢,dut bound.Invent. Math.121 (1995), #1, 211–233.
A. Garcia and H. Stichtenoth: On the asymptotic behaviour of some towers of function fields over finite fields.J. Number Theory61 (1996), #2, 248–273.
A. Garcia and H. Stichtenoth: Asymptotically good towers of function fields over finite fields.C. R. Acad. Sci. Paris 1322 (1996), #11, 1067–1070.
A. Garcia, H. Stichtenoth and M. Thomas: On towers and composita of towers of function fields over finite fields.Finite Fields and their Appl.3 (1997), #3, 257–274.
E.-U. Gekeler: Drinfeld-Moduln und modulare Formen iiber rationalen Funktionenkörpern. Bonner Math. Schriften 119, 1980.
E.-U. Gekeler:Drinfeld Modular Curves.Berlin: Springer-Verlag, 1980 (Lecture Notes in Math. 1231).
E.-U. Gekeler: Über Drinfeld’sche Modulkurven vom Hecke-Typ.Compositio Math.57 (1986), #2, 219–236.
E.-U. Gekeler and U. Nonnengardt: Fundamental domains of some arithmetic groups over function fields.International J. Math.6 (1995), #5, 689–708.
V. D. Goppa: Codes on algebraic curves.Soviet Math. Dokl.24 (1981), #1, 170–172.
D. Goss: 0x70-adic Eisenstein series for function fields.Compositio Math. 41(1980), #1, 3–38.
Y. Hamahata: Tensor products of Drinfeld modules and v-adic representations.Man’usc. Math. 79(1993), #3–4, 307–327.
Y. Ihara: Congruence relations and Shimura curves. Pages 291–311 ofAutomorphic Forms Representations,and L-functions (A. Borel and W. Casselman, eds.; Providence: AMS, 1979; Part 2 of Vol. 33 of Proceedings of Symposia in Pure Mathematics).
Y. Ihara: Some remarks on the number of rational points of algebraic curves over finite fields.J. Fac. Sci. Tokyo 28(1981), #3, 721–724.
M. A. Tsfasman and S. G. Vlá dut:Algebraic-Geometric Codes.Dordrecht: Kluwer, 1991.
M. A. Tsfasman, S. G. V1á,dutt and T. Zink: Modular curves, Shimura curves and Goppa codes better than the Varshamov-Gilbert bound.Math. Nachr. 109(1982), 21–28.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2001 Springer Basel AG
About this paper
Cite this paper
Elkies, N.D. (2001). Explicit Towers of Drinfeld Modular Curves. In: Casacuberta, C., Miró-Roig, R.M., Verdera, J., Xambó-Descamps, S. (eds) European Congress of Mathematics. Progress in Mathematics, vol 202. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8266-8_14
Download citation
DOI: https://doi.org/10.1007/978-3-0348-8266-8_14
Publisher Name: Birkhäuser, Basel
Print ISBN: 978-3-0348-9496-8
Online ISBN: 978-3-0348-8266-8
eBook Packages: Springer Book Archive