Questions tagged [elliptic-curves]

For questions regarding elliptic curves. Questions on ellipses should be tagged [conic-sections] instead.

Filter by
Sorted by
Tagged with
9
votes
3answers
5k views

Integral points on an elliptic curve

Let's start with an elliptic curve in the form $$E : y^2 = x^3 + Ax + B, \qquad A, B \in \mathbb{Z}.$$ I am wondering about integral points. I know that Siegel proved that $E$ has only finitely many ...
35
votes
1answer
4k views

How to compute rational or integer points on elliptic curves

This is an attempt to get someone to write a canonical answer, as discussed in this meta thread. We often have people come to us asking for solutions to a diophantine equation which, after some clever ...
29
votes
3answers
8k views

Find integer in the form: $\frac{a}{b+c} + \frac{b}{c+a} + \frac{c}{a+b}$

Let $a,b,c \in \mathbb N$ find integer in the form: $$I=\frac{a}{b+c} + \frac{b}{c+a} + \frac{c} {a+b}$$ Using Nesbitt's inequality: $I \ge \frac 32$ I am trying to prove $I \le 2$ to implies there ...
34
votes
1answer
2k views

Does an elementary solution exist to $x^2+1=y^3$?

Prove that there are no positive integer solutions to $$x^2+1=y^3$$ This problem is easy if you apply Catalans conjecture and still doable talking about Gaussian integers and UFD's. However, can this ...
10
votes
2answers
2k views

Calculating the divisors of the coordinate functions on an elliptic curve

I am currently reading Silverman's arithmetic of elliptic curves. In chapter II, reviewing divisor, there is an explicit calculation: Given $y^2 = (x-e_1)(x-e_2)(x-e_3)$ let $P_i = (e_i,0),$ and $ P_\...
17
votes
2answers
733 views

The term “elliptic”

There are many things which are called “elliptic” in various branches of mathematics: Elliptic curves Elliptic functions Elliptic geometry Elliptic hyperboloid Elliptic integral Elliptic modulus ...
3
votes
2answers
249 views

On $p^2 + nq^2 = z^2,\;p^2 - nq^2 = t^2$ and the “congruent number problem”

(Much revised for brevity.) An integer $n$ is a congruent number if there are rationals $a,b,c$ such that, $$a^2+b^2 = c^2\\ \tfrac{1}{2}ab = n$$ or, alternatively, the elliptic curve, $$y = x^3-n^...
26
votes
2answers
8k views

Elliptic Curves and Points at Infinity

My undergraduate number theory class decided to dip into a bit of algebraic geometry to finish up the semester. I'm having trouble understanding this bit of information that the instructor presented ...
19
votes
1answer
830 views

More elliptic curves for $x^4+y^4+z^4 = 1$?

(Note: This has been updated to be similar with this MO post.) There are exactly 31 known primitive solutions to, $$a^4+b^4+c^4 = d^4\tag{1}$$ with $d<10^{28}$. Noam Elkies showed that $(1)$ as, ...
13
votes
3answers
2k views

References for elliptic curves

I just finished reading Silverman and Tate's Rational Points on Elliptic Curves and thought it was very interesting. Could any of you point me to some more references (ex. books, articles) on ...
11
votes
2answers
1k views

Making an elliptic curve out of a cubic polynomial made a cube, or $ax^3+bx^2+cx+d = y^3$

What is the transformation such that a general cubic polynomial to be made a cube, $$ax^3+bx^2+cx+d = y^3\tag{1}$$ can be transformed to Weierstrass form, $$x^3+Ax+B = t^2\tag{2}$$ (The special ...
18
votes
2answers
519 views

Exploiting a Diophantine approximation of $\pi^4$ into giving a series of rationals for $\pi^4$

A note about this question: The original question asked seems likely impossible so I am really asking if we can exploit the technique below into giving us a 'nice' form for $\pi^4$. By nice form I ...
11
votes
3answers
2k views

Integer solutions for $x^3+2=y^2$?

I've heard a famous result that $26$ is the only integer, such that $26-1=25$ is a square number and $26+1=27$ is a cubic number.In other words, $(x,y)=(5,3)$ is the only solution for $x^2+2=y^3$. ...
4
votes
2answers
415 views

Elliptic curve $y^2= x^3 + x$ over the finite field $\mathbb{F}_p$ with $p \geq 3$.

Consider the elliptic curve $$E: y^2= x^3 + x$$ over the finite field $\mathbb{F}_p$ with $p \geq 3$. I want to show that $|E(\mathbb{F}_p)| \equiv 0 \mod 4$. I know that, if $p \equiv 3\mod 4$, ...
10
votes
2answers
1k views

Local-Global Principle and the Cassels statement.

In a recent article I have read, i.e. " Lecture notes on elliptic curves ", Prof.Cassels remarks in page-110 that There is not merely a local-global principle for curves of genus-$0$, but it ...
6
votes
1answer
558 views

transform into weierstrass-form

How can I transform the elliptic curve $E/\mathbb{C}$ of the form $$y^2=4(x-e_1)(x-e_2)(x-e_3)$$ with $e_1>e_2>e_3\in\mathbb{R}$ roots of $E$ into a Weierstrass-Form $y^2=x^3+ax+b$?
24
votes
4answers
681 views

On the integral $\int_0^1\frac{dx}{\sqrt[4]x\ \sqrt{1-x}\ \sqrt[4]{1-x\,\gamma^2}}=\frac{1}{N}\,\frac{2\pi}{\sqrt{2\gamma}}$

V. Reshetnikov gave the interesting integral, $$\int_0^1\frac{\mathrm dx}{\sqrt[4]x\ \sqrt{1-x}\ \sqrt[4]{2-x\,\sqrt3}}=\frac{2\,\sqrt2}{3\,\sqrt[8]3}\pi\tag1$$ After some experimentation, it turns ...
13
votes
3answers
2k views

Can you recommend some books on elliptic function?

I plan to study elliptic function. Can you recommend some books? What is the relationship between elliptic function and elliptic curve?Many thanks in advance!
12
votes
2answers
3k views

The modular curve X(N)

I have a question about the modular curve X(N), which classifies elliptic curves with full level N structure. (A level N structure of an elliptic curve E is an isomorphism from $Z/NZ \times Z/NZ$ to ...
8
votes
2answers
821 views

Birational Equivalence of Diophantine Equations and Elliptic Curves

A while ago I saw this question Quartic diophantine equation: $16r^4+112r^3+200r^2-112r+16=s^2$ which was very relevant to a undergraduate research paper I am currently working on. The answer given ...
8
votes
3answers
201 views

For which $n$ can $(a, nb, c)$ and $(b, c, d)$ be Pythagorean triples?

Fermat proved that if $(a, b, c)$ is a Pythagorean triple, then $(b, c, d)$ cannot be a Pythagorean triple. Suppose $(a, nb, c)$ form a Pythagorean triple. Can $(b, c, d)$ be a Pythagorean triple? ...
4
votes
1answer
1k views

Cubic diophantine equation

How can be solved the equation $x^3+x-1=y^2$ in positive integers? I know this equation defines an elliptic curve, but this seems to be a non-elementary way to solve this question. Is there a more ...
4
votes
2answers
3k views

Finding integer solutions to $y^2=x^3-2$

I have the equation: $$y^2=x^3-2$$ It seems to be deceivingly simple, yet I simply cannot crack it. It is obviously equivalent to finding a perfect cube that is two more than a perfect square, and a ...
2
votes
1answer
212 views

Using Modularity Theorem and Ribet's Theorem to disprove existence of rational solutions

This is likely overly optimistic, but can one take the results from the Modularity theorem and Ribet's theorem, and distill these down to an undergrad math level of a way to check if certain rational ...
4
votes
2answers
245 views

The group $E(\mathbb{F}_p)$ has exactly $p+1$ elements

Let $E/\mathbb{F}_p$ the elliptic curve $y^2=x^3+Ax$. We suppose that $p \geq 7$ and $p \equiv 3 \pmod {4}$. I want to show that the group $E(\mathbb{F}_p)$ has exactly $p+1$ elements. I was ...
4
votes
1answer
304 views

Finding two non-congruent right-angle triangles

The map $g: B \to A, \ (x,y) \mapsto \left(\dfrac {x^2 - 25} y, \dfrac {10x} y, \dfrac {x^2 + 25} y \right)$ is a bijection where $A = \{ (a,b,c) \in \Bbb Q ^3 : a^2 + b^2 = c^2, \ ab = 10 \}$ and $B =...
1
vote
2answers
112 views

Finding all rational points in $x^2+y^2=6$.

I want to find all rational points in the circle $x^2+y^2=6$. This would be easy if I could find one rational point in the circle, however, it's very hard to guess one in this case. However, I don't ...
0
votes
2answers
401 views

Prove that there are $p+1$ points on the elliptic curve $y^2 = x^3 + 1$ over $\mathbb{F}_p$, where $p > 3$ is a prime such that $p \equiv 2 \pmod 3$.

Let $p > 3$ be a prime such that $p \equiv 2 \pmod 3$. Define the elliptic curve $E$ over $\mathbb{F}_p$ by $y^2 = x^3 + 1$. Prove that $E(\mathbb{F}_p)$ consists of $p+1$ points. Using Fermat's ...
0
votes
1answer
179 views

Elliptic Curve and Divisor Example help (Step 1)

I have an elliptic curve $E$ over $\mathbb{F}_{11}$ defined by $y^2=x^3+4x$ with the point at infinity $\mathcal{O}$ I have a divisor of $E$, defined by $$D=\left[(0,0)\right]+\left[(2,4)\right]+\...
0
votes
1answer
106 views

Elliptic Curve and Divisor Example help (Step 3) [duplicate]

I am reading this paper, specifically Example 2.3 on page 9, and am having a few problems understanding a part of it We construct an elliptic curve $E$ on $\mathbb{F}_{11}$ defined by $y^2=x^3+4x$ ...
8
votes
3answers
3k views

Group Law for an Elliptic curve

I was reading this book "Rational points on Elliptic curves" by J.Silverman, and J.Tate, 2 prominent figures in Number theory and was very intrigued after reading the first couple of pages. The ...
23
votes
2answers
4k views

Explicit Derivation of Weierstrass Normal Form for Cubic Curve

In page 22-23 of Rational Points on Elliptic Curves by Silverman and Tate, authors explain why is it possible to put every cubic curve into Weierstrass Normal Form. Here are relevant pages: (My ...
17
votes
3answers
4k views

History of elliptic curves

In one sense elliptic curves are a rather modern object as some of its properties have been studied only in the last century or so. But in another sense there are a very classical object for studying ...
12
votes
1answer
788 views

Is the pushforward of the sheaf of differentials on an elliptic curve over a scheme necessarily trivial?

If $f:E\rightarrow S$ is an elliptic curve over a scheme $S$ (so $f$ is proper and smooth of relative dimension one with geometrically connected fibers of genus one, equipped with a section $0:S\...
7
votes
2answers
201 views

An elliptic curve $x(x - 78126^2)(x - 4\times5^7) = y^2$ for $x_1^7+x_2^7+\dots +x_8^7= x_9^7$

Ajai Choudhry found an infinite number of primitive solutions to, $$x_1^7+x_2^7+\dots +x_8^7= x_9^7$$ by using an elliptic curve with a parameter $m=2$. There are variants, the one I used is, $$x(x ...
7
votes
4answers
1k views

Proving $x^4+y^4=z^2$ has no integer solutions

I need solution check to see if I overlooked something: If $x^4+y^4=z^2$ has an integer solution then $(\frac{x}{y})^4+1=(\frac{z}{y^2})^2$ has a solution in rationals. Second equation is ...
10
votes
2answers
768 views

A question on FLT and Taniyama Shimura

Sometime back i watched the documentary of Andrew Wiles proving the Fermat's Last theorem. A truly inspiring video and i still watch it whenever i am in a depressed mood. There are certain things(...
5
votes
1answer
847 views

Intuition and Stumbling blocks in proving the finiteness of WC group

After reading many articles about the Tate-Shafarevich Group ,i understood that "in naive perspective the group is nothing but the measure of the failure of Hasse principle, and coming to its ...
9
votes
2answers
3k views

Rational solutions of $y^2 = x^3 - x$

I believe that the only rational solutions of $$y^2 = x^3 - x$$ are the obvious ones $(-1,0)$, $(0,0)$, $(1,0)$, and that this was proved by Fermat using the method of descent. Can anyone outline a ...
7
votes
1answer
725 views

solving $x^3-2y^3=1$ using cubic number field

I am trying to solve the diophantine equation $x^3-2y^3=1$ using $\mathbb{Q}(\sqrt[3]{2}).$ I've read this link: Solve $x^3 +1 = 2y^3$ The following is what i have tried: Finding all integer ...
6
votes
3answers
3k views

How do you determine if an elliptic curve over a finite field is cyclic?

I know the group order and the points of the elliptic curve $y^2 = x^3 + Ax + B$, but I am confused on how to determine if they from a cyclic group The curve $y^2 = x^3 + 2x +2$ in $\Bbb F_{11}$ ...
9
votes
0answers
510 views

Generalizing Ramanujan's cube roots of cubic roots identities

(This extends this post.) Define the function, $$\sqrt[3]{G(t)} = \sqrt[3]{t+x_1}+\sqrt[3]{t+x_2}+\sqrt[3]{t+x_3}\tag1$$ where the $x_i$ are roots of the cubic, $$x^3+ax^2+bx+c=0\tag2$$ While $G(t)...
7
votes
1answer
1k views

On Bachet's Duplication Formula and the number $-432$

While reading "Rational Points on Elliptic Curves" by Silverman and Tate, I came across this interesting passage about Bachet's duplication formula: I know how to derive Bachet's duplication formula ...
6
votes
1answer
268 views

Rational map of a curve to an elliptic curve

If I have a curve given by $$ y^2 = (x^3-1)(x^3-a), $$ how do I find out if there is a rational variable transformation $y=y(s,t)$, $x=x(s,t)$ that maps this curve onto an elliptic curve of the form $$...
6
votes
1answer
2k views

Find all integer solutions to $x^2+4=y^3$. [duplicate]

Find all integer solutions to $x^2+4=y^3$. Some obvious solutions are $(x,y)=(\pm2,2)$. Are these the only ones?
6
votes
2answers
344 views

Book recomendation for Elliptic-curve cryptography

I have already taken a course on Cryptography, The course focused mainly on the public-key cryptography based on the algebraic structure of elliptic curves over finite fields. Now, I would like to ...
8
votes
1answer
193 views

Why does the elliptic curve for $a+b+c = abc = 6$ involve a solvable nonic?

The curve discussed in this OP's post, $$\color{brown}{-24a+36a^2-12a^3+a^4}=z^2\tag1$$ is birationally equivalent to an elliptic curve. Following E. Delanoy's post, let $G$ be the set of rational ...
5
votes
1answer
2k views

How to find all rational points on the elliptic curves like $y^2=x^3-2$

Reading the book by Diophantus, one may be led to consider the curves like: $y^2=x^3+1$, $y^2=x^3-1$, $y^2=x^3-2$, the first two of which are easy (after calculating some eight curves to be solved ...
5
votes
1answer
228 views

Cube of an integer

$\frac{x}{y}+\frac{y}{z}+\frac{z}{x}=k$ and $x, y, z, k$ are integers. Prove that $xyz$ is cube of some integer number. I was wondering about giving a parametrization for the rational points on the ...
4
votes
2answers
566 views

Rational points on an elliptic curve

Consider the following elliptic curve $y^2=(x+1540)(x-508)(x-65024)$. It is trivial that the points $P_1(-1540,0)$, $P_2(508,0)$ and $P_3(65024,0)$ lie on this curve. It is also quite easy to find ...