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## 数学代写|椭圆曲线代考Elliptic Curves代考|Math395 Projective Space

$$y^2=x^3+a x+b$$

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## 数学代写|椭圆曲线代考Elliptic Curves代考|CM Elliptic Curve

Slightly more generally, we start with $\mathbb{G}_m$ over a $p$-adic integral local domain $B$ whose quotient field is $K$. This means that we regard the functor $\mathbb{G}_m$ as defined over the category of $B$-algebras (changing the base ring appears to be a trivial maneuver but has a strong impact on the structure of the corresponding scheme; see Sect. 4.1).

Take an algebraic closure $\bar{K}$ of $K$. We suppose that a prime $l$ is invertible in $B$ and $\mu_{l \infty}(\bar{K}) \subset B$ (that is, all $l$-power roots of unity in $\bar{K}$ are contained in $B)$. Fix a prime $l$ prime to the characteristic of $K$. Let $K\left(\mathbb{G}m\right)=K(t)$ be the rational function field of $\mathbb{G}_m=\operatorname{Spec}\left(B\left[t, t^{-1}\right]\right)$ [i.e., $K(t)$ is made up of $\frac{P(t)}{Q(t)}$ for all polynomials $\left.0 \neq Q(t), P(t) \in B[t]\right]$. Define a linear operator $U(l)$ acting on rational functions $\phi \in K(t)$ on $\mathbb{G}_m$ by $$\phi \mid U(l)(t)=\frac{1}{l} \sum{\zeta \in \mu_l} \phi\left(\zeta t^{1 / l}\right)$$
We call this operator the Hecke operator of the prime $l$.

The Hecke operator has the following geometric interpretation: We have the $l$-power group homomorphism $[l]: \mathbb{G}_m \rightarrow \mathbb{G}_m$ given by $x \mapsto x^l$. The map $[l]$ is actually an endomorphism of the functor $\mathbb{G}_m$. In this sense, it is essentially surjective $\left(\mathbb{G}_m\right.$ is a divisible group under fppf topology over $B$ in the geometric terminology in Sect. 4.1.10), as $s \in \mathbb{G}_m(A)$ is in the image of $[l]$ from $\mathbb{G}_m(A[\sqrt[l]{a}])$. Its kernel is the group $\mu_l$ of $l$ th roots of unity whose affine ring is a smaller ring $B[t] /\left(t^l-1\right)$ free of finite rank $l$ over $B$ (we call the rank $l$ the order of $\mu_l$; so $\mu_l$ has finite “order” $l$ ).

For the moment, take $B$ to be a field $K$. Then the affine ring $K\left[t, t^{-1}\right]$ of $\mathbb{G}{m / K}$ has infinite dimension as a $K$-vector space and has Krull dimension 1 as a ring, hence far bigger than the affine ring $K[t] /\left(t^l-1\right)$ of $\mu{N / K}$ which has dimension $l$ as $K$-vector space and Krull dimension 0 as a ring. In this sense, $\mu_N$ is “small” (see [CRT] Sect. 5 for Krull dimension).

Return to a general base ring $B$. Since $[l]$ is essentially surjective with small “finite” kernel, we call $[l]$ an isogeny. We will see later that $[l]$ is an étale isogeny if $l$ is invertible in $B$ (see Sect. 4.1.4 for the exact notion of étale map, but here “étale” basically means that for any maximal ideal $\mathfrak{m}$ of $B, B / \mathfrak{m}\left[\mu_l\right]$ is a separable extension of $B / \mathrm{m}$; see [CRT] Sect. 26 for separability). Then taking $t$ to be the variable on the target $\mathbb{G}_m$, we have $t^{1 / l}$ in the source $\mathbb{G}_m$. Then $\phi \circ[l]^{-1}=\phi\left(t^{1 / l}\right)$ is an element of the function field of the source $\mathbb{G}_m$, which is a degree- $l$ Kummer extension of the function field of the target $\mathbb{G}_m$.

## 数学代写|椭圆曲线代考Elliptic Curves代考|Invariant Differential Operators

Shimura studied the effect on modular forms of the following invariant differential operators on the upper half-complex plane $\mathfrak{H}$ indexed by $k \in \mathbb{Z}$ :
$\delta_k=\frac{1}{2 \pi \sqrt{-1}}\left(\frac{\partial}{\partial z}+\frac{k}{2 y \sqrt{-1}}\right)$ for $z=x+i y \in \mathfrak{H}$ and $\delta_k^r=\delta_{k+2 r-2} \cdots \delta_k$,
where $r \in \mathbb{Z}$ with $r \geq 0$. For some other properties of these operators not stated here, see [LFE] 10.1. Here are easy identities:
Exercise 1.26 Show the following formulas:

1. $\delta_{k+l}(f g)=g \delta_k f+f \delta_l g$.
2. For a holomorphic function $f: \mathfrak{H} \rightarrow \mathbb{C}, \delta_k^r\left(\left.f\right|k \alpha\right)=\left.\left(\delta_k^r f\right)\right|{k+2 r} \alpha$ if $\left.f\right|_k \alpha(z)=\operatorname{det}(\alpha)^{k / 2} f(\alpha(z))(c z+d)^{-k}$ for $\alpha=\left(\begin{array}{ll}a & b \ c & d\end{array}\right) \in G L_2(\mathbb{R})$ with $\operatorname{det}(\alpha)>0$

By the exercise, if $f \in G_k\left(\Gamma_1(N) ; \mathbb{C}\right), \delta_k^r(f)$ satisfies $\left.\delta_k^r(f)\right|_{k+2 r} \gamma=\delta_k^r(f)$ for all $\gamma \in \Gamma_1(N)$. Although $\delta_k^r(f)$ is not a holomorphic function, defining
$$\delta_k^r(f)(w)=w_2^{-k-2 r} \delta_k^r(f)(z),$$
we have a well-defined homogeneous modular form. In this sense, $\delta_k^r(f)$ is a real-analytic modular form on $\Gamma_1(N)$ of weight $k+2 r$.

## 数学代写|椭圆曲线代考Elliptic Curves代考|CM Elliptic Curve

$$\phi \mid U(l)(t)=\frac{1}{l} \sum \zeta \in \mu_l \phi\left(\zeta t^{1 / l}\right)$$

Hecke 算子有以下几何解释: 我们有 $l$-龺群同态 $[l]: \mathbb{G}m \rightarrow \mathbb{G}_m$ 由 $x \mapsto x^l$. 地图 $[l]$ 实际上是函子的自同态 $\mathbb{G} m$. 从这个意义上 说，它本质上是满射的 ( $\mathbb{G}_m$ 是 fppf 拓扑下的可分群 $B$ 在 Sect. 的几何术语中。4.1.10)，作为 $s \in \mathbb{G}_m(A)$ 在图像中 $[l]$ 从 $\mathbb{G}_m(A[\sqrt[l]{a}])$. 它的内核是群 $\mu_l$ 的 $l$ 仿射环是较小环的单位根 $B[t] /\left(t^l-1\right)$ 不受有限等级限制 $l$ 超过 $B$ (我们称排名 $l$ 的顺序 $\mu_l$; 所 以 $\mu_l$ 有有限的”秩序” $l$ ). 目前，采取 $B$ 成为一个领域 $K$. 然后是仿射环 $K\left[t, t^{-1}\right]$ 的 $\mathrm{G} m / K$ 具有无限维度作为 $K$-向量空间并且具有 $\mathrm{Krull}$ 维度 1 作为环， 因此远大于仿射环 $K[t] /\left(t^l-1\right)$ 的 $\mu N / K$ 有维度 $l$ 作为 $K$-向量空间和 Krull 维度 0 作为环。在这个意义上， $\mu_N$ 是”小的”（请参 阅 [CRT] 第 5 节了解 Krull 维度)。 返回通用基环 $B$. 自从 $[l]$ 本质上是小“有限”内核的满射，我们称 $[l]$ 同种异体。我们稍后会看到 $[l]$ 是 etale 同源如果 $l$ 是可逆的 $B$ (有 关 étale 映射的确切概念，请参阅第 4.1.4 节，但这里的“étale”基本上意味着对于任何最大理想 $\mathfrak{m}$ 的 $B, B / \mathfrak{m}\left[\mu_l\right]$ 是的可分离扩展 函数域的一个元倩 $\mathbb{G}_m$ ，这是一个学位 $l l$ 目标函数域的 Kummer 扩展 $\mathbb{G}_m$.

## 数学他写|椭圆曲线代考Elliptic Curves代考|Invariant Differential Operators Shimura

1. $\delta_{k+l}(f g)=g \delta_k f+f \delta_l g$.
2. 对于全纯函数 $f: \mathfrak{H} \rightarrow \mathbb{C}, \delta_k^r(f \mid k \alpha)=\left(\delta_k^r f\right) \mid k+2 r \alpha$ 如果 $\left.f\right|k \alpha(z)=\operatorname{det}(\alpha)^{k / 2} f(\alpha(z))(c z+d)^{-k{\text {为了 }}}$ $\alpha=\left(\begin{array}{lll}a & b c & d\end{array}\right) \in G L_2(\mathbb{R})$ 和det $(\alpha)>0$
通过练习，如果 $f \in G_k\left(\Gamma_1(N) ; \mathbb{C}\right), \delta_k^r(f)$ 满足 $\left.\delta_k^r(f)\right|_{k+2 r} \gamma=\delta_k^r(f)$ 对所有人 $\gamma \in \Gamma_1(N)$. 虽然 $\delta_k^r(f)$ 不是全纯函数，定义
$$\delta_k^r(f)(w)=w_2^{-k-2 r} \delta_k^r(f)(z),$$
我们有一个明确定义的同质模块化形式。在这个意义上， $\delta_k^r(f)$ 是 个实解析模形式 $\Gamma_1(N)$ 重量 $k+2 r$.

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

Posted on Categories:Elliptic Curves, 数学代写, 数论, 椭圆曲线

## 数学代写|椭圆曲线代考Elliptic Curves代考|MATH7304 Hecke Operators for Gm

$$y^2=x^3+a x+b$$

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## 数学代写|椭圆曲线代考Elliptic Curves代考|How Many Integer Points?

Let $C$ be a non-singular cubic curve given by an equation
$$a x^3+b x^2 y+c x y^2+d y^3+e x^2+f x y+g y^2+h x+i y+j=0$$
with integer coefficients. We have seen that if $C$ has a rational point (possibly at infinity), then the set of all rational points on $C$ forms a finitely generated abelian group. So we can get every rational point on $C$ by starting from some finite set and adding points using the geometrically defined group law.

Another natural number theoretic problem is that of describing the solutions $(x, y)$ to the cubic equation with $x$ and $y$ both integers. Since the cubic equation may have infinitely many rational points, we are asking which of those rational points have integer coordinates.
For a curve given by a Weierstrass equation
$$C: y^2=x^3+a x^2+b x+c,$$
the Nagell-Lutz theorem tells us that points of finite order have integer coordinates. It is natural to ask if the converse is true. A little experimentation shows that it is not. We saw one example in Section 4.3, where we showed

that the curve $y^2=x^3+3$ has no points of finite order, but it clearly has the integer point $(1,2)$. Similarly, it is easy to show that the curve $y^2=x^3+17$ has no points of finite order, yet it has lots of integer points, including
$$(-2, \pm 3), \quad(-1, \pm 4), \quad(2, \pm 5), \quad(4, \pm 9), \quad(8, \pm 23),$$
and six other points that we leave as an exercise for you to discover.

## 数学代写|椭圆曲线代考Elliptic Curves代考|Taxicabs and Sums of Two Cubes

The title of this section may provoke some curiosity since it is the first time in the book that we have referred to methods of conveyance. The reference has to do with a famous mathematical story. When the brilliant Indian mathematician Ramanujan was in the hospital in London, his colleague G.H. Hardy came to visit. Hardy remarked that he had come in taxicab number 1729 , and surely that was a rather dull number. Ramanujan instantly replied that, to the contrary, 1729 is a very interesting number. It is the smallest number expressible as a sum of two cubes in two different ways. Thus
$$1729=9^3+10^3=1^3+12^3 .$$
So the taxicab number 1729 gives a cubic curve
$$x^3+y^3=1729$$
that has two integer points. Of course, we can switch $x$ and $y$, so we end up with four points,
$$(9,10), \quad(10,9), \quad(1,12), \quad(12,1) .$$
We claim that there are no other integer points. This is a special case of Siegel’s theorem (Theorem 5.1), but in this case the proof is easy because the cubic $x^3+y^3$ factors.

## 数学代写|椭圆曲线代考Elliptic Curves代考|How Many Integer Points?

$$a x^3+b x^2 y+c x y^2+d y^3+e x^2+f x y+g y^2+h x+i y+j=0$$

$$C: y^2=x^3+a x^2+b x+c,$$
Nagell-Lutz 定理告诉伐们有限阶的点具有整数坐标。很自然地会问反过来是否成立。一点实验表明它不是。我们在第 $4.3$ 节中看 到了一个例子，我们展示了

$$(-2, \pm 3), \quad(-1, \pm 4), \quad(2, \pm 5), \quad(4, \pm 9), \quad(8, \pm 23),$$

## 数学代写椭圆曲线代考Elliptic Curves代考|Taxicabs and Sums of Two Cubes

$$1729=9^3+10^3=1^3+12^3 .$$

$$x^3+y^3=1729$$

$$(9,10), \quad(10,9), \quad(1,12), \quad(12,1) .$$

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

Posted on Categories:Elliptic Curves, 数学代写, 数论, 椭圆曲线

## 数学代写|椭圆曲线代考Elliptic Curves代考|MAT322-01 Mordell’s Theorem

$$y^2=x^3+a x+b$$

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## 数学代写|椭圆曲线代考Elliptic Curves代考|The Discriminant

After our digression into real and complex analysis, we return to the field of rational numbers. As always, we take our curve in its normal form
$$y^2=f(x)=x^3+a x^2+b x+c$$
where $a, b, c$ are rational numbers. If we let $X=d^2 x$ and $Y=d^3 y$, then our equation becomes
$$Y^2=X^3+d^2 a X^2+d^4 b X+d^6 c$$
By choosing a large integer $d$, we can clear any denominators in $a, b$, and $c$. So from now on we will assume that our cubic curve is given by an equation having integer coefficients.

Our goal in this chapter is to prove a theorem, first proven (independently) by Nagell and Lutz in the 1930s, which will tell us how to find all of the rational points of finite order. Their theorem has two parts. The first part says that if $(x, y)$ is a rational point of finite order, then its coordinates are integers. The second part says that either $y=0$, in which case it is a point of order two, or else $y \mid D$, where $D$ is the discriminant of the polynomial $f(x)$. In particular, a cubic curve has only a finite number of rational points of finite order.

## 数学代写|椭圆曲线代考Elliptic Curves代考|Points of Finite Order Have Integer Coordinates

Now we come to the most interesting part of the Nagell-Lutz theorem, the proof that a rational point $(x, y)$ of finite order must have integer coordinates. We will show that $x$ and $y$ are integers in a rather indirect way. We observe that one way to show that a positive integer equals 1 is to show that it is not divisible by any primes. Thus we can break the problem up into an infinite number of subproblems, namely we show that when the rational numbers $x$ and $y$ are written in lowest terms, their denominators are not divisible by 2 , and they are not divisible by 3 , and they are not divisible by 5 , and so on.
So we let $p$ be some prime, and we try to show that $p$ does not divide the denominator of $x$ and does not divide the denominator of $y$. That leads us to consider the set of rational points $(x, y)$ where $p$ does divide the denominator of $x$ or $y$.

It will be helpful to set some notation. Every non-zero rational number may be written uniquely in the form $\frac{m}{n} p^\nu$, where $m$ and $n$ are integers that are prime to $p$ and with $n \geq 1$ and where the fraction $m / n$ is in lowest terms. We define the order of such a rational number to be the exponent $\nu$, and we write
$$\operatorname{ord}\left(\frac{m}{n} p^\nu\right)=\nu$$
To say that $p$ divides the denominator of a rational number is the same as saying that its order is negative, and similarly to say that $p$ divides the numerator of a rational number is the same as saying that its order is positive. The order of a rational number is zero if and only if $p$ divides neither its numerator nor its denominator.

## 数学代写椭圆曲线代考Elliptic Curves代考|The Discriminant

$$y^2=f(x)=x^3+a x^2+b x+c$$

$$Y^2=X^3+d^2 a X^2+d^4 b X+d^6 c$$

## 数学代写|椭圆曲线代考Elliptic Curves代考|Points of Finite Order Have Integer Coordinates

$$\operatorname{ord}\left(\frac{m}{n} p^\nu\right)=\nu$$

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

Posted on Categories:Elliptic Curves, 数学代写, 数论, 椭圆曲线

## 数学代写|椭圆曲线代考Elliptic Curves代考|MAT4240 Rational Points on Conics

$$y^2=x^3+a x+b$$

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。