Question

Let \(C\) be an irreducible cubic, \(O\) a simple point on \(C\) giving rise to the addition \(\oplus\) on the set \(C^{\circ}\) of simple points. Suppose another \(O^{\prime}\) gives rise to an addition \(\oplus^{\prime}\). Let \(Q=\varphi\left(O, O^{\prime}\right)\), and define \(\alpha:(C, O, \oplus) \rightarrow\left(C, O^{\prime}, \oplus^{\prime}\right)\) by \(\alpha(P)=\varphi(Q, P) .\) Show that \(\alpha\) is a group isomorphism. So the structure of the group is independent of the choice of \(O .\)

Short answer

Question: Prove that the structure of an abelian group formed using the chord-and-tangent addition law on a cubic curve is independent of the choice of the origin. Answer: To prove that the structure of the abelian group is independent of the choice of the origin, we showed that there exists a group isomorphism between two groups with different origins. Specifically, we demonstrated that a map α between the two groups is both a bijection and preserves the group operation. This guarantees that the structure of the group is independent of the choice of the origin.

Step-by-step solution

Show that \(\alpha\) is a bijection

To prove that \(\alpha\) is a bijection, we need to show that it is both injective (one-to-one) and surjective (onto). Injective: Assume \(\alpha(P_1) = \alpha(P_2)\) for some \(P_1, P_2 \in C^{\circ}\). Then, \(\varphi(Q, P_1) = \varphi(Q, P_2)\). According to the properties of the addition law, this implies that \(P_1 = P_2\). Thus, \(\alpha\) is injective. Surjective: Let \(R \in C^{\circ}\) be an arbitrary point. We need to find a point \(P \in C^{\circ}\) such that \(\alpha(P) = R\). Since \(\varphi\) is a bijective map, there exists a unique point \(P\) such that \(Q \oplus P = R\). Then, \(\alpha(P) = \varphi(Q, P) = R\). Thus, \(\alpha\) is surjective. Since \(\alpha\) is both injective and surjective, it is a bijection.

Show that \(\alpha\) preserves the group operation

Now, we need to show that for all \(A, B \in C^{\circ}\), \(\alpha(A \oplus B) = \alpha(A) \oplus^{\prime} \alpha(B)\). Let \(A, B \in C^{\circ}\). Then, $$ \begin{aligned} \alpha(A \oplus B) &= \varphi(Q, A \oplus B) \\ &= \varphi(Q \oplus A, B) \\ &= \varphi(\varphi(Q, A), B) \\ &= \varphi(\alpha(A), B) \\ &= \alpha(A) \oplus B \\ &= \alpha(A) \oplus^{\prime} \alpha(B). \end{aligned} $$ Hence, \(\alpha\) preserves the group operation. Since \(\alpha\) is a bijection and preserves the group operation, it is a group isomorphism, and therefore the structure of the group is independent of the choice of the origin \(O\).

Key concepts

Algebraic Curves

Algebraic curves are fundamental concepts in geometry and algebra, particularly within the field known as algebraic geometry. These curves are defined as the sets of solutions to polynomial equations in two variables, plotted in a coordinate space. More precisely, a curve can be defined by an equation of the form \( f(x, y) = 0 \), where \( f(x, y) \) is a polynomial with coefficients in a field, such as the field of real or complex numbers.

Understanding algebraic curves is crucial because they serve as examples for more complex algebraic varieties. They also offer a visual and tangible representation of solutions to polynomial equations, making the abstract notions in algebra more accessible. Many classical problems in mathematics, such as finding tangent lines, studying points at infinity, and classifying singularities, can be approached by examining the properties of algebraic curves. In our context, a cubic, an algebraic curve defined by a polynomial equation of degree three, is the main subject of study.

Cubic Curves

Cubic curves are a special class of algebraic curves that are defined by polynomial equations of degree three. The general form of a cubic curve in the Cartesian plane is given by: \( ax^3 + bx^2y + cxy^2 + dy^3 + ex^2 + fxy + gy^2 + hx + iy + j = 0 \), where \( a \) through \( j \) are constants.

Some cubic curves are well-known, such as the elliptic curves, which have profound implications in number theory and cryptography. One interesting feature of cubic curves is their potential to form a group structure under a certain geometrically defined addition operation. This operation is intimately connected with the properties of the curve and can often be visualized by drawing lines intersecting the curve. For instance, the addition of two points might be defined by the third intersection point of a line through the two points with the curve. The versatility and complexity of cubic curves make them a rich area of study in algebraic geometry.

Group Theory in Algebraic Geometry

Group theory provides a powerful algebraic framework to study the symmetries and structure-preserving transformations of mathematical objects. In the realm of algebraic geometry, group theory plays a pivotal role in understanding the algebraic structures that can be formed by geometric objects, such as algebraic curves.

In this context, the term 'group' refers to a collection of elements that can be combined using an operation that satisfies certain axioms: closure, associativity, having an identity element, and each element having an inverse. For algebraic curves like cubic curves, the points can form a group with respect to a geometrically defined addition operation. Such groups are not only abstract constructs; their properties can sometimes be visualized in the way points on the curve combine and relate to each other. This deep connection between algebraic structures and geometry is what makes group theory so useful in algebraic geometry.

Group Operations

The concept of group operations is at the heart of group theory. A group operation is a rule for combining two elements of a group to produce another element within the same group. In the context of algebraic curves, the idea of using points on a curve to form a group involves defining such a group operation.

For example, on elliptic curves, the group operation can be illustrated using a geometric approach: if we take two points \(A\) and \(B\) on the curve, the line that passes through these points will intersect the curve at a third point \(C\). The point opposite to \(C\), reflected across the x-axis, is defined as the sum of \(A\) and \(B\) under this operation. Such operations must adhere to the axioms of group theory, such as having an identity element, which here is often assumed to be a 'point at infinity' acting as a neutral element for the addition. Understanding these operations is vital in proving properties such as the existence of group isomorphisms between structures defined by different base points on the curve, as in the exercise provided.