Characterization of the torsion of the Jacobians of two families of hyperelliptic curves

摘要

Using the classical Nagell-Lutz Theorem, Mazur's deep result, and the reduction modulo primes homomorphism, it is relatively easy to calculate the torsion of any given elliptic curve over Q. This calculation may be slighty more complicated for infinite (one-parameter) families of such curves. Among them the families E~a: y~2 = x~3+ax and E_b: y~2 = x~3+b occupy a special place (without loss of generality we can and will assume that a and b are nonzero integers 4th and 6th power free respectively). Their respective j -invariants are 1728 and 0. Both families have complex multiplication by a fourth and third root of unity respectively. For E = E~a or E_b, let E(Q)_(tors) denote the torsion subgroup of the Mordell-Weil group E(Q). The following results are well known (see for instance [K, Theorems 5.2, 5.3, p. 134].