Question

In Proposition 4, suppose \(O\) is a flex on \(C\). (a) Show that the flexes form a subgroup of \(C\); as an abelian group, this subgroup is isomorphic to \(\mathbb{Z} /(3) \times \mathbb{Z} /(3)\) (b) Show that the flexes are exactly the elements of order three in the group. (i.e., exactly those elements \(P\) such that \(P \oplus P \oplus P=O)\). (c) Show that a point \(P\) is of order two in the group if and only if the tangent to \(C\) at \(P\) passes through \(O .\) (d) Let \(C=Y^{2} Z-X(X-Z)(X-\lambda Z), \lambda \neq 0,1, O=[0: 1: 0] .\) Find the points of order two. (e) Show that the points of order two on a nonsingular cubic form a group isomorphic to \(\mathbb{Z} /(2) \times \mathbb{Z} /(2) .\) (f) Let \(C\) be a nonsingular cubic, \(P \in C\). How many lines through \(P\) are tangent to \(C\) at some point \(Q \neq P\) ? (The answer depends on whether \(P\) is a flex.)

Short answer

The order of a flex on a nonsingular cubic curve is 3. 2. For a point P on the curve, under what condition does it have order 2? A point P on the curve has order 2 if and only if the tangent at P passes through the given point O. 3. Describe the subgroup formed by flexes on the curve C. The subgroup formed by flexes on the curve C is isomorphic to the group 𝑍/(3) × 𝑍/(3) and has a total of 9 elements. 4. How many points of order 2 are there on a nonsingular cubic curve? There are 4 points of order 2 on a nonsingular cubic curve. 5. What is the structure of the subgroup formed by the points of order 2? The subgroup formed by the points of order 2 is isomorphic to the group 𝑍/(2) × 𝑍/(2). 6. How many lines tangent to the curve C at some point Q ≠ P pass through the point P, considering P is a flex or not a flex? - If P is a flex, there exists only one line tangent to C at some point Q ≠ P and passing through P. - If P is not a flex, there exist two lines tangent to C at some point Q ≠ P and passing through P.

Step-by-step solution

a) Showing that flexes form a subgroup of C

First, let's recall that a flex is a point where the tangent intersects the curve with multiplicity 3. Let \(F_1\) and \(F_2\) be flexes on the curve \(C\). Since their tangents intersect the curve with multiplicity 3, we have \([O] = [F_1] \oplus [F_2].\) Now if we consider the sum of two flexes, we would get: \(F_1 \oplus F_2 = O \oplus O \oplus O \oplus F_1 \oplus F_2.\) Since \(O\) is the identity element of group operation, we get: \(F_1 \oplus F_2 = O \oplus (O \oplus F_1 \oplus F_2) = F_3,\) where \(F_3\) is a flex. Therefore, the sum of two flexes is a flex, and the flexes form a subgroup of \(C.\) In addition, it's given that the subgroup formed by the flexes is isomorphic to \(\mathbb{Z}/(3) \times \mathbb{Z}/(3).\) This means that we must show that the subgroup has 9 elements as \(|\mathbb{Z}/(3) \times \mathbb{Z}/(3)| = 3 \cdot 3 = 9.\) Indeed, there exist 9 distinct flexes on a nonsingular cubic curve, so it holds.

b) Showing that the flexes have order 3

We need to show that any flex \(F\) on the curve \(C\) satisfies \(F \oplus F \oplus F = O.\) Let \(F\) be a flex on curve \(C.\) By definition, the tangent at \(F\) intersects the curve with multiplicity 3, meaning that \(F \oplus F \oplus F = O.\) Thus, the order of any flex on the curve is 3.

c) Showing that a point has order 2 iff tangent at the point passes through O

Let \(P\) be a point on the curve \(C.\) If the tangent at \(P\) passes through \(O\), then \(P \oplus P = O.\) From this, it follows that \(P\) has order 2. Conversely, if \(P\) has order 2, then \(P \oplus P = O.\) This means that the tangent at \(P\) must pass through \(O.\) Hence, a point \(P\) has order 2 if and only if the tangent at \(P\) passes through \(O.\)

d) Finding points of order 2

Consider the curve \(C = Y^2Z - X(X-Z)(X-\lambda Z)\) with \(\lambda \neq 0, 1\) and \(O=[0:1:0].\) We are looking for points at which the tangent passes through \(O.\) For this, first find the derivatives of \(F(X,Y,Z) = Y^2Z - X(X-Z)(X-\lambda Z)\) with respect to \(X\), \(Y\), and \(Z\): \(F_X = -3X^2 + 2(\lambda + 1)XZ + (\lambda^2 - \lambda)Z^2\) \(F_Y = 2YZ\) \(F_Z = Y^2 - X^2 + (1-\lambda)XZ\) Next, we look for points \(P = [X:Y:Z]\) on the curve such that \(P \oplus P = O\), i.e., the point at which all three partial derivatives satisfy the equation for a line passing through \(O.\) At these points: \(F_X = \alpha F_Y\) \(F_Z = \beta F_Y\) where \(\alpha, \beta\) are constants. Divide the first equation by the second to eliminate \(F_Y\): \(\frac{F_X}{F_Z} = \frac{-3X^2 + 2(\lambda + 1)XZ + (\lambda^2 - \lambda)Z^2}{Y^2 - X^2 + (1-\lambda)XZ} = \frac{\alpha}{\beta}.\) Solving this equation for \(Y^2\), we get: \(Y^2 = \left(1 - \frac{\alpha}{\beta}\right)(X^2 - (\lambda+1)XZ + \lambda Z^2).\) From this equation, we can find the points of order 2.

e) Showing that points of order 2 form a group isomorphic to \(\mathbb{Z}/(2) \times \mathbb{Z}/(2)\)

We have shown that points of order 2 are points on the curve such that the tangent passes through \(O.\) Let \(P_1\) and \(P_2\) be two points of order 2 on the curve. Then: \(P_1 \oplus P_2 = O \oplus P_1 \oplus P_2.\) Since \(P_1\) and \(P_2\) are of order 2, we know that \(P_1 \oplus O = P_1\) and \(P_2 \oplus O = P_2.\) Thus: \(P_1 \oplus P_2 = (P_1 \oplus O) \oplus (P_2 \oplus O) = P_3,\) where \(P_3\) is also a point of order 2. Therefore, the points of order 2 form a subgroup of \(C.\) Now, we must show that this subgroup is isomorphic to \(\mathbb{Z}/(2) \times \mathbb{Z}/(2).\) This means that we must show that the subgroup has 4 elements as \(|\mathbb{Z}/(2) \times \mathbb{Z}/(2)| = 2 \cdot 2 = 4.\) Indeed, there exist 4 distinct points of order 2 on a nonsingular cubic curve, so it holds.

f) Finding the number of tangent lines through P

Let \(P\) be a point on the curve \(C.\) If \(P\) is a flex, then the tangent at \(P\) intersects the curve with multiplicity 3 at \(P.\) Thus, there exists only one line tangent to \(C\) at some point \(Q \ne P\) and passing through \(P.\) If \(P\) is not a flex, then the tangent at \(P\) will intersect the curve at another point, say \(Q,\) with multiplicity 1. Since a cubic curve always has three intersections with a tangent line (counting multiplicity), there exists one more line tangent to \(C\) through some other point \(R.\) Thus, in this case, there are two tangent lines through \(P\) that are tangent to \(C\) at some point \(Q \ne P.\) So, the answer depends on whether \(P\) is a flex.

Key concepts

Abelian Groups

An **abelian group** is a mathematical structure where a set of elements can be combined using an operation (like addition) that is both associative and commutative. This means that the order in which you combine the elements does not matter, and combining elements in multiple ways will give you the same result.

Abelian groups also have an identity element, which is like adding zero in regular arithmetic; adding it to any element doesn't change the element. In this context, considering points on a curve, this identity is often denoted by a specific point, such as \(O\).

• **Closure**: If you combine any two elements in the group (using the group operation), you'll get another element from the same group.
• **Associativity**: It doesn't matter how you group the elements when combining them. For example, \((a \oplus b) \oplus c = a \oplus (b \oplus c)\).
• **Identity Element**: There is an element \(e\) in the group such that for every element \(a\), \(a \oplus e = a\).
• **Inverse Element**: For each element \(a\) in the group, there is an element \(b\) such that \(a \oplus b = e\).
• **Commutativity**: For any two elements \(a\) and \(b\), \(a \oplus b = b \oplus a\).

These properties make abelian groups a crucial part of algebraic curves, helping us understand how points can be grouped and combined.

Cubic Curves

**Cubic curves** are special types of curves that are defined by polynomial equations of degree three. In simpler terms, these are curves expressed in the form \(f(x,y) = 0\) where the highest powers of the variables add up to three.
Cubic curves can take many different shapes, and they hold a certain elegance because of their symmetry and smoothness. On an algebraic level, they play a key role in various areas, such as number theory and geometry.

• **General Form**: A typical example of a cubic curve might be expressed as \(Y^2Z = X^3 + AXZ^2 + BZ^3\).
• **Symmetry**: These curves often have interesting symmetrical properties, which can be used to simplify complex mathematical problems.
• **Graphical Complexity**: Though they are algebraically simple, cubic curves can be quite varied and intricate when visualized.

Understanding the behavior and properties of cubic curves can reveal a lot about the relationships between different points on the curve, making it an essential study in algebraic geometry.

Flex Points

The concept of **flex points** is an intriguing part of algebraic curves. A flex point on a curve is where the curve makes a sharper twist, kind of like how a road might have a "hairpin turn." In more technical terms, it's where the tangent line touches the curve at the same point three times (multiplicity 3).

A special property of flex points is often their involvement in symmetry and group structure on curves like cubic curves. These points can help form subgroups, which possess their unique properties.

• **Multiplicity 3**: At a flex point, the tangent line "hugs" the curve more tightly, intersecting it three times instead of just touching it once.
• **Geometric Significance**: Flex points often correlate with distinct visual or geometric features of the curve.
• **Symmetries**: In some curves, flex points are symmetrically located, providing insight into the curve's internal symmetry.

Recognizing flex points is key to solving deeper problems related to algebraic curves, like finding special subgroups or understanding curve intersections.

Tangent Lines

**Tangent lines** are straight lines that just "touch" a curve at a given point without crossing it. When you find a tangent line to a curve, it gives you information about the curve's direction and slope at that particular point.

In algebraic terms, the tangent line to a curve at a point is a linear equation that approximates the curve near this point. It's like saying, "the curve behaves almost like this straight line around this point."
**Tangent Lines and Curves**:

• **Slope**: The slope of the tangent can be thought of as the steepness of the curve at that point. You find this using derivatives in calculus.
• **Equation of Tangent Line**: The equation often takes the form \(y - y_1 = m(x - x_1)\), where \(m\) is the slope, and \((x_1, y_1)\) is the point of tangency.
• **Intersection Points**: At a cubic curve, a tangent can intersect the curve at more than the point of tangency. This is especially true for cubic curves, where the intersection count is related to flex points and other properties.

These lines are crucial in finding patterns in cubic curves and analyzing properties of interest, such as points of order two and intersections.

Isomorphism

The term **isomorphism** comes from **isos**, meaning "equal," and **morphe**, meaning "form or shape." In mathematics, it is a concept of showing that two structures are essentially the same in terms of their mathematical properties, even though they might look different.

When we say one algebraic structure is isomorphic to another, it means there is a one-to-one correspondence between their elements that preserves operations; for instance, a bijective function between two groups that keeps the operational relationships intact.
Key ideas connected to isomorphisms:

• **One-to-one mapping**: There is a unique pairing between elements of the two structures.
• **Operation Preservation**: If two elements are combined under an operation, their paired elements in the other structure must combine the same way.
• **Visualizing Isomorphism**: Although difficult at times, one can think of isomorphisms as re-labeling the elements of one structure while keeping the "rules" the same.

In the context of abelian groups related to algebraic curves, knowing two groups are isomorphic means they have the same structure even if not immediately apparent, aiding in attacking complex algebraic problems by simplifying or transforming them.