Maximal curves and Tate-Shafarevich results for quartic and sextic twists |
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Affiliation: | 1. Bernoulli Institute for Mathematics, Computer Science, and Artificial Intelligence, Nijenborgh 9, 9747 AG Groningen, the Netherlands;2. University of Campinas (UNICAMP), Institute of Mathematics, Statistics and Computer Science (IMECC), Rua Sérgio Buarque de Holanda, 651, Cidade Universitária, 13083-859, Campinas, SP, Brazil |
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Abstract: | We study elliptic surfaces corresponding to an equation of the specific type , defined over the finite field for a prime power . It is shown that if defines a curve that is maximal over then the rank of the group of sections defined over on the elliptic surface is determined in terms of elementary properties of the rational function . Similar results are shown for elliptic surfaces given by using prime powers and curves . Finally, for each of the forms used here, existence of curves with the property that they are maximal over is discussed, as well as various examples. |
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Keywords: | Finite field Maximal curve Function field Elliptic curve Elliptic surface Mordell-Weil rank |
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