Prove The Lower Bound Of The Canonical Height Of Points On The Elliptic Curve E D : Y 2 = X 3 + A D 2 X + B D 3 E_{d}:y^2=x^3+Ad^2x+Bd^3 E D ​ : Y 2 = X 3 + A D 2 X + B D 3

Introduction

In the realm of arithmetic geometry, elliptic curves play a pivotal role in understanding the behavior of rational points on algebraic curves. The canonical height of a point on an elliptic curve is a fundamental concept that measures the complexity of the point's coordinates. In this article, we will delve into the problem of proving the lower bound of the canonical height of points on the elliptic curve $E_{d}:y^2=x^3+Ad^2x+Bd^3$. This problem is an exercise from Silverman's "The Arithmetic of Elliptic Curves" and requires a deep understanding of the properties of elliptic curves and the concept of canonical height.

Background and Notation

Before we proceed, let's establish some notation and background information. We are given a nonzero integer $A$ and $B$ such that $4A^3+27B^2\neq 0$. This condition ensures that the elliptic curve $E_{d}$ is non-singular. For each integer $d\neq 0$, we define the elliptic curve $E_{d}/\mathbb{Q}$ as:

$$E_{d}:y^2=x^3+Ad^2x+Bd^3$$

The canonical height of a point $P$ on an elliptic curve $E$ is denoted by $\hat{h}(P)$ and is defined as:

$$\hat{h}(P)=\lim_{n\to\infty}\frac{1}{2^n}h(2^nP)$$

where $h(P)$ is the Weil height of $P$.

The Lower Bound of the Canonical Height

We are tasked with proving the lower bound of the canonical height of points on the elliptic curve $E_{d}$. To do this, we need to establish a lower bound for the Weil height of points on $E_{d}$. We can use the following inequality:

$$h(P)\geq\frac{1}{2}\log\left(\max\left\{|x(P)|,|y(P)|\right\}\right)$$

where $P=(x(P),y(P))$ is a point on $E_{d}$.

Proof of the Lower Bound

To prove the lower bound of the canonical height, we need to show that:

$$\hat{h}(P)\geq\frac{1}{2}\log\left(\max\left\{|x(P)|,|y(P)|\right\}\right)$$

for all points $P$ on $E_{d}$.

Let's consider a point $P=(x,y)$ on $E_{d}$. We can write the equation of the elliptic curve as:

$$y^2=x^3+Ad^2x+Bd^3$$

Using the inequality above, we can write:

$$h(P)\geq\frac{1}{2}\log\left(\max\left\{|x|,|y|\right\}\right)$$

Now, let's consider the point $2P=(2x,2y)$. We can write the equation of the elliptic curve as:

$$(2y)^2=(2x)^3+Ad^2(2x)+Bd^3$$

Simplifying the equation, we get:

$$4y^2=8x^3+4Ad^2x+2Bd^3$$

Using the inequality above, we can write:

$$h(2P)\geq\frac{1}{2}\log\left(\max\left\{|2x|,|2y|\right\}\right)$$

Now, let's consider the point $2^nP=(2^n x,2^n y)$. We can write the equation of the elliptic curve as:

$$(2^n y)^2=(2^n x)^3+Ad^2(2^n x)+Bd^3$$

Simplifying the equation, we get:

$$2^{2n}y^2=2^{3n}x^3+2^nAd^2x+2^nBd^3$$

Using the inequality above, we can write:

$$h(2^nP)\geq\frac{1}{2}\log\left(\max\left\{|2^n x|,|2^n y|\right\}\right)$$

Now, let's take the limit as $n\to\infty$:

$$\lim_{n\to\infty}\frac{1}{2^n}h(2^nP)\geq\lim_{n\to\infty}\frac{1}{2}\log\left(\max\left\{|2^n x|,|2^n y|\right\}\right)$$

Using the properties of logarithms, we can simplify the right-hand side:

$$\lim_{n\to\infty}\frac{1}{2^n}h(2^nP)\geq\frac{1}{2}\log\left(\max\left\{|x|,|y|\right\}\right)$$

This shows that:

$$\hat{h}(P)\geq\frac{1}{2}\log\left(\max\left\{|x|,|y|\right\}\right)$$

for all points $P$ on $E_{d}$.

Conclusion

In this article, we have proven the lower bound of the canonical height of points on the elliptic curve $E_{d}:y^2=x^3+Ad^2x+Bd^3$. We have shown that the canonical height of a point $P$ on $E_{d}$ is bounded below by:

$$\hat{h}(P)\geq\frac{1}{2}\log\left(\max\left\{|x(P)|,|y(P)|\right\}\right)$$

This result has important implications for the study of elliptic curves and their rational points. We hope that this article has provided a clear and concise proof of the lower bound of the canonical height of points on elliptic curves.

References

• Silverman, J. H. (2009). The Arithmetic of Elliptic Curves. Springer-Verlag.
• Lang, S. (1999). Elliptic Curves: Diophantine Analysis. Springer-Verlag.
• Cassels, J. W. S. (1991). Lectures on Elliptic Curves. Cambridge University Press.
  Q&A: Proving the Lower Bound of the Canonical Height of Points on Elliptic Curves ====================================================================================

Introduction

In our previous article, we proved the lower bound of the canonical height of points on the elliptic curve $E_{d}:y^2=x^3+Ad^2x+Bd^3$. In this article, we will address some common questions and concerns that readers may have regarding this result.

Q: What is the significance of the lower bound of the canonical height?

A: The lower bound of the canonical height is a fundamental concept in arithmetic geometry that measures the complexity of the coordinates of a point on an elliptic curve. It has important implications for the study of elliptic curves and their rational points.

Q: How does the lower bound of the canonical height relate to the Weil height?

A: The lower bound of the canonical height is closely related to the Weil height. In fact, we used the inequality $h(P)\geq\frac{1}{2}\log\left(\max\left\{|x(P)|,|y(P)|\right\}\right)$ to prove the lower bound of the canonical height.

Q: What is the relationship between the canonical height and the height of a point on an elliptic curve?

A: The canonical height of a point $P$ on an elliptic curve $E$ is defined as $\hat{h}(P)=\lim_{n\to\infty}\frac{1}{2^n}h(2^nP)$. This means that the canonical height is a limiting value of the height of a point on the elliptic curve.

Q: How does the lower bound of the canonical height affect the study of elliptic curves?

A: The lower bound of the canonical height has important implications for the study of elliptic curves and their rational points. It provides a lower bound for the complexity of the coordinates of a point on an elliptic curve, which is essential for understanding the behavior of rational points on elliptic curves.

Q: Can you provide an example of how the lower bound of the canonical height is used in practice?

A: Yes, the lower bound of the canonical height is used in various applications, such as:

• Cryptographic protocols: The lower bound of the canonical height is used in cryptographic protocols, such as the elliptic curve digital signature algorithm (ECDSA), to ensure the security of the protocol.
• Number theory: The lower bound of the canonical height is used in number theory to study the properties of elliptic curves and their rational points.
• Algebraic geometry: The lower bound of the canonical height is used in algebraic geometry to study the properties of algebraic curves and their rational points.

Q: What are some common mistakes to avoid when working with the lower bound of the canonical height?

A: Some common mistakes to avoid when working with the lower bound of the canonical height include:

• Incorrectly applying the inequality: Make sure to apply the inequality correctly and avoid making mistakes when simplifying the expression.
• Failing to consider the limiting value: Make sure to consider the limiting value of the height a point on the elliptic curve when calculating the canonical height.
• Ignoring the relationship between the canonical height and the Weil height: Make sure to understand the relationship between the canonical height and the Weil height and avoid making mistakes when working with these concepts.

Conclusion

In this article, we have addressed some common questions and concerns that readers may have regarding the lower bound of the canonical height of points on elliptic curves. We hope that this article has provided a clear and concise explanation of this result and its implications for the study of elliptic curves and their rational points.

References

• Silverman, J. H. (2009). The Arithmetic of Elliptic Curves. Springer-Verlag.
• Lang, S. (1999). Elliptic Curves: Diophantine Analysis. Springer-Verlag.
• Cassels, J. W. S. (1991). Lectures on Elliptic Curves. Cambridge University Press.