We investigate equations of the form x³ + y² + bx + a = 0, where a and b are numbers. The solution space of an equation on this form is a curve in the xy-plane. See "What is Algebraic Geometry?" This curve has a certain shape, and we may wonder if other values of a and b give curves of the same shape. Instead of just trying arbitrary a's and b's and comparing the resulting curves, we use a theorem of algebraic geometry which says that the a and b values that give similar curves are a's and b's which themselves lie on a specific curve in the ab-plane.

Given an equation, for example x³ - y² = 0, we can plot all solutions to this equation to get the curve in Figure 1. The solutions (0,0), (1,1), and (1,-1) are marked as examples of points on the curve. All subsequent plots will be done to the same scale, but the numbers will be left out.

 

Figure 1: The algebraic curve of x³ - y² = 0.

Remark. The reader who is familiar with complex numbers will see that we get complex solutions for negative x-values. Due to the obvious difficulty of drawing complex numbers we restrict ourselves to drawing only real solutions.

The most important task in algebraic geometry is classification. That is, given a set of curves we want to group similar curves. We interpret "similar" to mean curves of the same shape. The Weierstrass family are curves given by equations of the form

x³ - y² + bx + a = 0

where a and b can vary. For example a = 0 and b = 0 gives x³ - y² = 0 as in Figure 1. Let us choose some different a's and b's and see what the corresponding curves look like. Figure 1 to Figure 4 show curves of different shape in the Weierstrass family. Given a and b, we want to be able to determine what kind of curve they will produce. To do this we must introduce a new curve, the discriminant, which is given by an equation of a and b.

 Figure 2: a = sqrt(4/27), b = -1.           Figure 3: a = b = 1.                   Figure 4: a = 0, b = 1.

 

Figure 5: The discriminant

The discriminant is defined as the solution to 27a² + 4b³ = 0. See Figure 5. The points in the figure are the a and b values that give the equations defining the curves in figure 1, 2, 3, and 4. For example, the point (1,1) (a = 1 and b = 1) in Figure 5 gives the equation x³ + y² + 1x + 1 = 0. We now have a curve which points correspond to curves in the Weierstrass family. If a curve crosses itself (Figure 2) or has an abrupt change (the (0,0) point in Figure 1 we say that the curve is singular. The curves in Figure 3 and 4 are non-singular. As these examples indicate, it is possible to show that curves with (a,b)- values above the discriminant are of the form of Figure 3. Curves with (a,b) values under the discriminant are of the form of Figure 4 and curves with (a,b) values on the discriminant are of the form of Figure 2 apart from (0,0) which is Figure 1.

Further, it is possible to show that two non-singular curves defined by (a₁,b₁) and (a₂,b₂) are similar if and only if j(a₁,b₁) = j(a₂,b₂), the "j-function" being defined as

j(a,b) = b³ / (27a² + 4b³).

As an arbitrary example, let us now see which a's and b's give j(a,b) = 1/8. The algebraic curve corresponding to

b³ / (27a² + 4b³) - 1/8 = 0.

The two arbitrary points on the curve in Figure 6 give the curves in Figure 7. In fact, it can be shown that every set of similar curves in the Weierstrass family are given by points (a,b) lying on a curve, like the one in Figure 6. In this way, all similar curves in the Weierstrass family are classified.

 

Figure 6: Two arbitrary points

Figure 7: The two curves that corresponds to the two points in Figure 6