Arithmetic on Normal Forms of Elliptic Curves
Explicit formulas for the 64 8-torsion points of the Tate Normal Form E4(b) are given that are valid over a field k of characteristic different from 2 containing a primitive eighth root of unity. The coordinates of the 4-torsion points are given by products of linear fractional expressions that are in terms of β, where β 4 = 16/(1-16b). The coordinates of the points of order 8 are shown to be products of algebraic expressions in β. These formulas are used to give a realization of the torsion subgroup Z8 ⊕ Z2 for infinitely many elliptic curves over Q in a different manner than given by Kubert. The coordinates of the 8-torsion points are shown to lie in an extension N(&zgr;16), where N is the normal closure of k(α) / k(j) and α is an indeterminate defined by the condition that (2/α,2/β) is a point on the quartic Fermat curve x 4 + y4 = 1. Formulas for the 36 6-torsion points of the Tate Normal Form E6(a) are given that are valid over a field k of characteristic different from 2 or 3 containing a primitive cube root of unity. The coordinates of the torsion points are written in terms of products of linear fractional expressions in α and β, where β 2 = α3 + 1 and the parameter a = (10β 2 – 18)/(9β2 – 9) = (10 α 3 – 8)/(9α3). It is shown that the torsion subgroup Z6 ⊕ Z6 is realized for infinitely many elliptic curves over a quartic extension of Q. In the case where E6(a) has complex mutiplication by the ring of integers of K = Q(√–d ) with d ≥ 2, d ≠ 3, d squarefree, the ray class field Σ6 of K is shown to equal K(ω,α,β) where ω is a primitive cube root of unity and α and β are related to a as above.
Morton, Purdue University.
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