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

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

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

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

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