Torsion points on the jacobian of a Fermat quotient

Let p≥5p≥5 be a prime, ζζ a primitive ppth root of unity and λ=1−ζλ=1−ζ. For 1≤s≤p−21≤s≤p−2, the smooth projective model Cp,sCp,s of the affine curve vp=us(1−u)vp=us(1−u) is a curve of genus (p−1)/2(p−1)/2 whose jacobian Jp,sJp,s has complex multiplication by the ring of integers of the cyclotomic field Q(ζ)Q(ζ). In 1981, Greenberg determined the field of rationality of the pp-torsion subgroup of Jp,sJp,s and moreover he proved that the λ3λ3-torsion points of Jp,sJp,s are all rational over Q(ζ)Q(ζ). In this paper we determine quite explicitly the λ3λ3-torsion points of Jp,1Jp,1 for p=5p=5 and p=7p=7, as well as some further pp-torsion points which have interesting arithmetical applications, notably to the complementary laws of Kummer’s reciprocity for ppth powers.