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Zeta functions of a class of elliptic curves
over a rational function field
of characteristic two

Authors: Ernst-Ulrich Gekeler, Rita Leitl and Bodo Wack
Journal: Math. Comp. 68 (1999), 823-833
MSC (1991): Primary :, 11G05, 11G40.; Secondary :, 11Y40
MathSciNet review: 1621527
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Abstract: We show how to calculate the zeta functions and the orders $|\Russian{X}|$ of Tate-Shafarevich groups of the elliptic curves with equation $Y^2+XY=X^3+\alpha X^2+\mbox{const}\cdot T^{-k}$ over the rational function field $\mathbf{F}_q(T)$, where $q$ is a power of 2. In the range $q=2$, $k \leq 37$, $\alpha \in \mathbf{F}_2\lbrack T^{-1}\rbrack$ odd of degree $\leq 19$, the largest values obtained for $|\Russian{X}|$ are $47^2$ (one case), $39^2$ (one case) and $27^2$ (three cases). We observe and discuss a remarkable pattern for the distributions of signs in the functional equation and of fudge factors at places of bad reduction. These imply strong restrictions on the precise form of the Langlands correspondence for GL$(2)$ over local or global fields of characteristic two.

References [Enhancements On Off] (What's this?)

  • [1] Bushnell, C., Fröhlich, A.: Gauss sums and $p$-adic division algebras. Lect. Notes Math. 987, Springer-Verlag 1983. MR 84m:12017
  • [2] Deligne, P.: Formes modulaires et représentations de GL(2). In Lect. Notes Math. 349, Springer-Verlag 1973, 55-105. MR 50:240
  • [3] Deligne, P.: Les constantes des équations fonctionnelles des fonctions $L$. In Lect. Notes Math. 349, Springer-Verlag 1973, 501-597. MR 50:2128
  • [4] Gekeler, E.-U.: Highly ramified pencils of elliptic curves in characteristic two. Duke Math. J. 89 (1997), 95-107. CMP 97:15
  • [5] Jacquet, H., Langlands, R.P.: Automorphic forms on GL(2). Lect. Notes Math. 114, Springer-Verlag 1970. MR 53:5481
  • [6] Leitl, R.: Elliptische Kurven über $\mathbf{F}_q(T)$ mit kleinem Führer, Diplomarbeit Saarbrücken 1995.
  • [7] Milne, J.S.: Arithmetic duality theorems. Academic Press, Boston-Orlando 1986. MR 88e:14028
  • [8] Serre, J.P.: Corps locaux, 2nd ed., Hermann, Paris 1968. MR 50:7096
  • [9] Shioda, T.: Mordell-Weil lattices and sphere packings, Am. J. Math. 113 (1991), 931-948. MR 92m:11066
  • [10] Shioda, T.: Some remarks on elliptic curves over function fields, Astérisque 209 (1992), 99-114. MR 94d:11046
  • [11] Tate, J.: On the conjectures of Birch and Swinnerton-Dyer and a geometric analog. In Dix exposés sur la cohomologie des schémas, North Holland, Amsterdam 1968. CMP 98:09; MR 39:2777
  • [12] Tate, J.: Algorithm for determining the type of a singular fiber in an elliptic pencil. In Lect. Notes Math. 476, Springer-Verlag 1975, 33-52. MR 52:13850
  • [13] Tunnell, J.: On the local Langlands conjecture for GL(2), Invent. Math. 46 (1978), 179-200. MR 57:16262

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Additional Information

Ernst-Ulrich Gekeler
Affiliation: Fachbereich 9 Mathematik, Universität des Saarlandes, Postfach 15 11 50, D-66041 Saarbrücken

Rita Leitl
Affiliation: Fachbereich 9 Mathematik, Universität des Saarlandes, Postfach 15 11 50, D-66041 Saarbrücken

Bodo Wack
Affiliation: Fachbereich 9 Mathematik, Universität des Saarlandes, Postfach 15 11 50, D-66041 Saarbrücken

Keywords: Elliptic curves, zeta functions, Tate-Shafarevich group, Langlands correspondence
Received by editor(s): August 30, 1996
Received by editor(s) in revised form: September 10, 1997
Additional Notes: Research supported by DFG, SP Algorithmische Zahlentheorie und Algebra.
Article copyright: © Copyright 1999 American Mathematical Society