Question

Sei \(L \in C\) ein Gitter mit der Bigenschaft \(g_{2}(L)=8\) und \(g_{s}(L)=0\). Der Punkt (2,4) liegt auf der affinen Kurve \(y^{2}=4 x^{3}-8 x .\) Sein Doppelted der in der elliptischen Kurve eingefuhrten Addition ist der Punkt \(\left(\frac{9}{4},-\frac{21}{4}\right)\), Anleitwng. Man bringe die Tangente an \((2,4)\) mit der Kurve zum Schnitt.

Short answer

The tangent at (2,4) intersects the curve at \(\left(\frac{9}{4}, -\frac{21}{4}\right)\).

Step-by-step solution

Verify the Given Curve Parameters

We start with the elliptic curve equation \(y^2 = 4x^3 - 8x\). Let's verify the coefficients related to the property. The given lattice properties \(g_2(L)=8\) and \(g_3(L)=0\) correspond to the Weierstrass form with terms having 0 for \(g_3(L)\). Validate these against the standard form \( y^2 = 4x^3 - g_2x - g_3 \), where \(a = -8\) and \(b = 0\). The curve is correct with respect to these parameters.

Find the Slope of the Tangent Line at (2,4)

To find the tangent line at the point \((2,4)\), calculate the derivative \(dy/dx\) using implicit differentiation of the curve equation \( y^2 = 4x^3 - 8x \). Differentiating gives: \(2y \cdot (dy/dx) = 12x^2 - 8\). Substitute \( (2,4) \) into the differentiated equation: \(2(4) \cdot (dy/dx) = 12(2)^2 - 8\), which simplifies to \(8(dy/dx) = 40\), leading to a slope of \(dy/dx = 5\). Thus, the slope of the tangent is 5.

Write the Equation of the Tangent Line

Using the point-slope form \( y - y_1 = m(x - x_1) \) with slope \(m = 5\) at point \((2, 4)\), the tangent line equation becomes \( y - 4 = 5(x - 2) \), which simplifies to \( y = 5x - 6 \).

Solve the Intersection of the Tangent with the Curve

Find the points of intersection between the tangent line and the curve by substituting the tangent line equation \(y = 5x - 6\) into the curve equation \(y^2 = 4x^3 - 8x\). This gives \((5x - 6)^2 = 4x^3 - 8x\). Expanding the left gives \(25x^2 - 60x + 36 = 4x^3 - 8x\). Rearrange to get a polynomial equation: \(4x^3 - 25x^2 + 52x - 36 = 0\).

Simplifying the Polynomial

Since \(x = 2\) is already a known solution (corresponds to point \((2,4)\)), use synthetic division or polynomial division to divide \(4x^3 - 25x^2 + 52x - 36\) by \(x-2\). The result is \(4x^2 - 17x + 18\). Factor or solve this quadratic using the quadratic formula to find the other intersection points.

Solving the Quadratic Equation

Solve \(4x^2 - 17x + 18 = 0\) using the quadratic formula: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) where \(a = 4, b = -17, c = 18\). Calculate the discriminant: \((-17)^2 - 4 \cdot 4 \cdot 18 = 1\). The roots are \(x = \frac{17 \pm 1}{8}\) which are \(x = 2.25\) and \(x = 2\). Since \((2,4)\) is the original point, \(x = 2.25\) gives the tangent intersection.

Verify Tangential Double Point

Since the problem statement provides the double point as \(\left(\frac{9}{4}, -\frac{21}{4}\right)\), confirm if \(x = 2.25\) matches. Substitute \(x = \frac{9}{4}\) back to solve for \(y\) using \(y = 5x - 6\). Substitute \(x\) to yield \(y = 5\times\frac{9}{4} - 6 = -\frac{21}{4}\). Hence, the point \(\left(\frac{9}{4}, -\frac{21}{4}\right)\) is correct and aligns with the given double point.

Key concepts

Weierstrass Equation

The Weierstrass Equation is a fundamental equation for elliptic curves, taking the form \( y^2 = 4x^3 - g_2x - g_3 \). This equation is pivotal since it offers a simplified method for analyzing complex elliptic curves using just two coefficients, \( g_2 \) and \( g_3 \). For any given elliptic curve, these coefficients help in defining its unique geometric properties. In the context of this exercise, the curve is given as \( y^2 = 4x^3 - 8x \). Here, \( g_2 = 8 \) and \( g_3 = 0 \), fitting it into the standard Weierstrass form. This particular setup specifies the lattice structure forming the basis for the elliptic curve, guiding us through further calculations.
Understanding the Weierstrass Equation is crucial when finding intersection points and properties of curves, as it becomes a starting point in many proofs and calculations surrounding elliptic curves.

Lattice Properties

Lattices, in the context of elliptic curves, refer to a grid of points that define the curve itself. These points are spread across the complex plane in a specific pattern, determined by the Weierstrass coefficients \( g_2 \) and \( g_3 \).
In this exercise, the lattice was specified with \( g_2 = 8 \) and \( g_3 = 0 \). These values are critical as they directly influence the shape and layout of the curve on the plane.
Lattice properties ensure that the elliptic curve behaves predictably, allowing for the application of operations such as point addition. Understanding these points and their spatial properties gives insight into how a curve can loop back upon itself and form an infinite structure with repeating patterns. This behavior is essential when considering problems involving the addition of points, such as determining the double of a given point on the curve.

Tangent Line

Finding the tangent line at a point on an elliptic curve is an important concept and involves calculus. To determine the tangent line at a specific point, such as \( (2,4) \) on the curve \( y^2 = 4x^3 - 8x \), we use implicit differentiation.
The derivative, or slope \( \frac{dy}{dx} \), is found by differentiating the curve equation and substituting the coordinates of the point. For our case, \( 2y \frac{dy}{dx} = 12x^2 - 8 \). Substituting \( (2,4) \) results in a slope \( \frac{dy}{dx} = 5 \).
The tangent line, using point-slope form, becomes \( y = 5x - 6 \). Understanding how to derive this line is crucial, as it helps in determining intersection points and performing operations on the elliptic curve.

Intersection Points

Intersection points between curves and lines are essential in studying elliptic curves. Here, we find intersection points by equating the equation of the tangent line \( y = 5x - 6 \) with the curve \( y^2 = 4x^3 - 8x \).
When substituting \( y = 5x - 6 \) into the curve equation, we solve \( (5x - 6)^2 = 4x^3 - 8x \) leading to a polynomial \( 4x^3 - 25x^2 + 52x - 36 = 0 \). Since \( x = 2 \) is a known root, we factor it out, simplifying to \( 4x^2 - 17x + 18 = 0 \).
The roots of this quadratic give us intersection points, \( x = 2 \) and \( x = 2.25 \). Solving for \( y \), we find the double point \( \left( \frac{9}{4}, -\frac{21}{4} \right) \), validating the earlier finding and showcasing typical interactions on elliptic curves.