Torsion points on the jacobian of a Fermat quotient |
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Authors: | R. Clement Ferná ndez |
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Affiliation: | a Departamento de Matemáticas, Facultad de Ciencia y Tecnología, Universidad del País Vasco, Apartado 644, 48080 Bilbao, Spain b Departamento de Economía Aplicada IV, Facultad de Ciencias Económicas y Empresariales, Universidad del País Vasco, Apartado 644, 48080 Bilbao, Spain |
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Abstract: | Let p≥5 be a prime, ζ a primitive pth root of unity and λ=1−ζ. For 1≤s≤p−2, the smooth projective model Cp,s of the affine curve vp=us(1−u) is a curve of genus (p−1)/2 whose jacobian Jp,s has complex multiplication by the ring of integers of the cyclotomic field Q(ζ). In 1981, Greenberg determined the field of rationality of the p-torsion subgroup of Jp,s and moreover he proved that the λ3-torsion points of Jp,s are all rational over Q(ζ). In this paper we determine quite explicitly the λ3-torsion points of Jp,1 for p=5 and p=7, as well as some further p-torsion points which have interesting arithmetical applications, notably to the complementary laws of Kummer’s reciprocity for pth powers. |
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Keywords: | 11G30 14G05 14G25 |
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