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Mathematics of Computation

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ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.98.

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Zeta functions of a class of elliptic curves over a rational function field of characteristic two
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by Ernst-Ulrich Gekeler, Rita Leitl and Bodo Wack PDF
Math. Comp. 68 (1999), 823-833 Request permission

Abstract:

We show how to calculate the zeta functions and the orders $|\Sha |$ 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 $|\Sha |$ 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
  • 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.
  • © Copyright 1999 American Mathematical Society
  • Journal: Math. Comp. 68 (1999), 823-833
  • MSC (1991): Primary :, 11G05, 11G40; Secondary :, 11Y40
  • DOI: https://doi.org/10.1090/S0025-5718-99-01058-3
  • MathSciNet review: 1621527