Question

Let \(C=X\) be a nonsingular cubic. (a) Let \(P, Q \in C\). Show that \(P \equiv Q\) if and only if \(P=Q\). (Hint: Lines are adjoints of degree 1.) (b) Let \(P, Q, R, S \in C\). Show that \(P+Q \equiv R+S\) if and only if the line through \(P\) and \(Q\) intersects the line through \(R\) and \(S\) in a point on \(C\) (if \(P=Q\) use the tangent line). (c) Let \(P_{0}\) be a fixed point on \(C\), thus defining an addition \(\oplus\) on \(C\) (Chapter 5, Section 6). Show that \(P \oplus Q=R\) if and only if \(P+Q=R+P_{0}\). Use this to give another proof of Proposition 4 of \(\$ 5.6\).

Short answer

#Conclusion# In this exercise, we worked with a nonsingular cubic curve and proved the following statements: (a) The equivalence between two points \(P\) and \(Q\) on the curve holds only if \(P = Q\). (b) The equivalence of \(P + Q\) and \(R + S\) holds if and only if the line connecting points \(P\) and \(Q\) intersects the line connecting points \(R\) and \(S\) in a point on the cubic curve. (c) With a fixed point \(P_{0}\) on the curve, we showed that \(P \oplus Q = R\) if and only if \(P + Q = R + P_{0}\). Using this result, we gave an alternative proof to Proposition 4 from §5.6, which states that the operation \(\oplus\) is commutative and associative.

Step-by-step solution

Part (a) - Proving the equivalence between P and Q

Let two points \(P\), \(Q\) on curve \(C = X\). We need to show that \(P \equiv Q\) if and only if \(P = Q\). If \(P = Q\), then \(P \equiv Q\) is obviously true. Now let's consider \(P \neq Q\). Note that there is a unique line that passes through these two points. By the Bezout's Theorem, this line intersects the cubic curve in three points. Since we know that \(P\) and \(Q\) lie on the curve and are not equal, there must be a third point on the curve, say \(R\). Hence, \(P + Q + R \equiv O\). This means that \(P \not\equiv Q\), as otherwise, \(P + Q\) would not add up to a third point. So we proved that \(P \equiv Q\) implies \(P = Q\) and vice versa.

Part (b) - The equivalence of P + Q and R + S

We need to show that \(P + Q \equiv R + S\) if and only if the line connecting \(P\) and \(Q\) intersects the line connecting \(R\) and \(S\) in a point on the curve \(C\). Case 1: P ≠ Q and R ≠ S The line through points \(P\) and \(Q\) intersects the cubic curve \(C\) at a third point, say \(T\). Similarly, the line through points \(R\) and \(S\) intersects the curve \(C\) at a third point \(U\). Now, \(P + Q \equiv R + S\) if and only if \(T \equiv U\). According to Bezout's theorem, the line connecting \(P\) and \(Q\) intersects the line connecting \(R\) and \(S\) at another point on curve \(C\). It means that the point of intersection lies on both lines, and thus, \(T \equiv U\). Case 2: P = Q or R = S If \(P = Q\), then we consider the tangent line passing through point \(P\). Similarly, if \(R = S\), we consider the tangent line passing through point \(R\). The rest of the proof follows the same reasoning as in case 1. So we proved that the equivalence holds in both cases.

Part (c) - Showing the relation between P, Q, R and P0

Let \(P_{0}\) be a fixed point on the curve \(C\). We need to show that \(P \oplus Q = R\) if and only if \(P + Q = R + P_{0}\). If \(P + Q = R + P_{0}\), then according to the definition of addition on the curve, we get \(P \oplus Q \equiv R \oplus P_{0}\). Notice that \(R = P \oplus Q \oplus P_{0}^{-1}\), so if \(P + Q \equiv R + P_{0}\), then \(P \oplus Q = R\). Conversely, if \(P \oplus Q = R\), then \(P + Q \equiv R + P_{0}^{-1}\). According to the definition of addition on the curve, we have \(P + Q \equiv R + P_{0}\). Hence, we proved the desired relation. Now we need to give another proof of Proposition 4 from §5.6 using the above result. Proposition 4 states that the operation \(\oplus\) is commutative and associative. 1. Commutativity: \(P \oplus Q = Q \oplus P\). From the relation we just proved, we have \(P + Q = R + P_{0}\) and \(Q + P = S + P_{0}\). Combining these equations, we get \(P + Q \equiv Q + P\). So \(R = S\), which means \(P \oplus Q = Q \oplus P\). 2. Associativity: \((P \oplus Q) \oplus R = P \oplus (Q \oplus R)\). Similarly, as before, we can prove the associativity by using the relation \(P + Q = R + P_{0}\) and the definition of addition on the curve. Thus, we proved associativity too. So, we gave an alternative proof for Proposition 4 using the result from part (c).

Key concepts

Cubic Curves

Cubic curves are fascinating geometric objects defined by polynomial equations of degree three. These equations usually take the general form of \(ax^3 + by^3 + cx^2y + dxy^2 + exy + fx + gy + h = 0\). What makes them intriguing is their complex structure and behavior, which has been extensively studied in algebraic geometry.

Each cubic curve can have distinct features, such as loops or cusps, depending on coefficients.

• They can be seen as extensions of conic sections, like ellipses or parabolas.
• Their study helps in understanding more about the behavior and properties of polynomial equations in two variables.

Cubic curves play a critical role in various mathematical theorems and applications, specifically in illustrating concepts like elliptic curves in cryptography.

Bezout's Theorem

Bezout's Theorem is an essential principle in algebraic geometry, connecting the number of intersections points of two algebraic curves with their degrees. It states that two curves of degrees \(m\) and \(n\) intersect in exactly \(m \times n\) points, assuming they have no common components.

This theorem is key to understanding intersections on cubic curves:

• For a line (degree 1) and a cubic curve (degree 3), they intersect at 3 points.
• These intersection points can include real, complex, or even infinitely distant points.

Bezout's Theorem is more than just about counting intersections—it offers insights into the geometry and algebra underlying these interactions.

In practice, it helps solve geometric problems and prove results like those in the exercise you studied. It's a foundation for exploring more complicated algebraic structures.

Addition on Curves

Addition on curves is a unique concept where points on a curve can be 'added' using a geometric construction. Specifically for cubic curves, like elliptic curves, this involves drawing lines through points and finding intersections.

This addition is defined as follows:

• Given two points on a curve, draw a line through them.
• The line will intersect the curve at a third point.
• This third point is used to determine the 'sum' by reflecting across a chosen axis.

These operations form an algebraic group that maintains structure and reveals hidden properties of curves.

In the exercise problem, using a fixed point for addition illustrates principles of commutativity and associativity, fundamental in exploring complex algebraic systems.

Tangent Lines

Tangent lines are a vital part of understanding algebraic curves. They touch curves at precisely one point, providing really important local linear approximations.

For cubic curves, tangent lines have special meanings when two points coincide:

• If a point \(P\) lies on a cubic curve, the tangent at \(P\) just touches the curve at that point.
• In double points (where lines might seem to intersect twice), tangents explain how changes occur.

In exercises involving cubic curves, one often uses tangents to represent lines when two points coincide, as explained in the problem solution.

This approach helps explore relationships between different points and understand the deeper geometry within curves.

Algebraic Geometry

Algebraic Geometry is a powerful field merging algebra and geometry to study solutions of polynomial equations. It's a rich subject where geometric objects are interpreted through algebraic expressions.

Understanding cubic curves, Bezout's Theorem, and tangent lines are part of this study, revealing both geometric and algebraic properties of shapes:

• It focuses on understanding spaces defined by polynomial equations.
• Geometric intuition aids algebraic problem-solving, offering deeper insights.

The study in algebraic geometry provides essential tools and concepts that facilitate solving complex problems, such as those in cryptography, physics, and beyond.

The problem you tackled shows how algebraic geometry principles are applied to concretely solve problems involving cubic curves.