By Dale Husemöller

First version offered over 2500 copies within the Americas; new version comprises 3 new chapters and new appendices

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One step in the proof is the demonstration that the index (E(k) : 2E(k)) is ﬁnite. It is clearly a necessary condition for E(k) to be ﬁnitely generated. , for Q to be in 2E(k). This theorem is proved using only the elementary methods of this chapter drawn from the classical theory of algebraic equations applied to analytic geometry. 1) Theorem. Let E be an elliptic curve deﬁned over a ﬁeld k by the equation y 2 = (x − α)(x − β)(x − γ ) = x 3 + ax 2 + bx + c, with α, β, γ ∈ k. For (x , y ) ∈ E(k) there exists (x, y) ∈ E(k) with 2(x, y) = (x , y ) if and only if x − α, x − β, and x − γ are squares.

This is a descent procedure which goes back to Fermat. In order to perform calculations with speciﬁc elliptic curves, it is convenient to put the cubic equation in a standard form. In 2 we show how, by changes of variable, we can eliminate three terms, y 3 , x y 2 , and wx 2 , from the ten-term general cubic equation given at the beginning of §4 and further normalize the coefﬁcients of x 3 and wy 2 to be one. The resulting equation is called an equation in normal form (or generalized Weierstrass equation) wy 2 + a1 wx y + a3 w 2 y = x 3 + a2 wx 2 + a4 w 2 x + a6 w 3 .

If x has degree 2 and y has degree 3 in the graded polynomial, then the equation has weight 6 when ai has weight i. The point at inﬁnity (0,0,1) is the zero of the group and the lines through this zero in the x, y-plane are exactly the vertical lines. This zero has the property that O O = O in terms of the chord-tangent composition so that three points add to zero in the elliptic curve if and only if they lie on a line in the plane of the cubic curve. In Chapter 1 we use this group law to calculate with an extensive number of examples.