2014
DOI: 10.3934/amc.2014.8.479
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Curves in characteristic $2$ with non-trivial $2$-torsion

Abstract: Cais, Ellenberg and Zureick-Brown recently observed that over finite fields of characteristic two, all sufficiently general smooth plane projective curves of a given odd degree admit a non-trivial rational 2-torsion point on their Jacobian. We extend their observation to curves given by Laurent polynomials with a fixed Newton polygon, provided that the polygon satisfies a certain combinatorial property. We also show that in each of these cases, the sufficiently general condition is implied by being ordinary. O… Show more

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Cited by 3 publications
(2 citation statements)
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“…One readily checks that the equation defining š¶ is smooth, and hence defines a hyperelliptic curve over F2 of genus g. Let š½ be the Jacobian of š¶. As in the proof of [14,Theorem 23], one sees that dim š½( F2 ) [2]…”
Section: The Results For General Curves: Statementmentioning
confidence: 90%
See 1 more Smart Citation
“…One readily checks that the equation defining š¶ is smooth, and hence defines a hyperelliptic curve over F2 of genus g. Let š½ be the Jacobian of š¶. As in the proof of [14,Theorem 23], one sees that dim š½( F2 ) [2]…”
Section: The Results For General Curves: Statementmentioning
confidence: 90%
“…Let J$J$ be the Jacobian of C$C$. As in the proof of [14, Theorem 23], one sees that prefixdimJ(trueFĀÆ2)[2]=g$\dim J(\bar{\mathbb {F}}_2)[2]=g$, and hence J$J$ is ordinary.ā–”$\Box$…”
Section: Residue Characteristicmentioning
confidence: 97%