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Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)

 

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 | References | Similar Articles | Additional Information

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.


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Additional Information

Ernst-Ulrich Gekeler
Affiliation: Fachbereich 9 Mathematik, Universität des Saarlandes, Postfach 15 11 50, D-66041 Saarbrücken
Email: gekeler@math.uni-sb.de

Rita Leitl
Affiliation: Fachbereich 9 Mathematik, Universität des Saarlandes, Postfach 15 11 50, D-66041 Saarbrücken
Email: rita@math.uni-sb.de

Bodo Wack
Affiliation: Fachbereich 9 Mathematik, Universität des Saarlandes, Postfach 15 11 50, D-66041 Saarbrücken
Email: bodo@math.uni-sb.de

DOI: http://dx.doi.org/10.1090/S0025-5718-99-01058-3
PII: S 0025-5718(99)01058-3
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