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

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6
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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 ...
25
votes
1answer
1k 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 ...
17
votes
2answers
2k 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 ...
4
votes
1answer
567 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 ...
4
votes
3answers
1k 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 ...
4
votes
3answers
1k 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 ...
4
votes
1answer
70 views

How I can express $(x,y)∈G$ by using the $r$ independent points $P_1,P_2,\ldots,P_r$

Let $C$ be an elliptic curve over $ℚ$. The group $C(ℚ)$ is a finitely generated Abelian group and we have $C(ℚ)≃ℤ^{r}⊕C(ℚ)^\mathrm{tors}$, where $C(ℚ)^\mathrm{tors}$ is a finite abelian group (is the ...
10
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0answers
263 views

Help with $x^4+y^4+z^4 = 1$?

There are exactly 20 known primitive solutions to, $$a^4+b^4+c^4 = d^4\tag{1}$$ with $d<10^{10}$. Noam Elkies (who kindly answered Question 1 below) showed that the form, $$(x+y)^4+(x-y)^4+z^4 = ...
2
votes
2answers
102 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 $ ...
6
votes
1answer
185 views

UPDATE: How to find the order of elliptic curve over finite field extension

I want to find the order of elliptic curve over the finite field extension $\mathbb{F}_{p^2}$, where $E(\mathbb{F}_{p^2}):y^2=x^3+ax+b $ I am using the method illustrated by John J. McGee in his ...
2
votes
1answer
43 views

What is the argument used to dsitinguish the cases (a) and (b)

We know from [B. Mazur, Modular curves and the Eisenstein ideal, Publ. math. IHES 47 (1977), 33-186] that if $C$ is an elliptic curve of the form ($C:y²=x³+ax+b$ with $a,b∈ℤ$), then $C(ℚ)^{tors}$ (the ...
3
votes
1answer
487 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 ...
2
votes
2answers
70 views

Reason behind standard names of coefficients in long Weierstrass equation

A long Weierstrass equation is an equation of the form $$y^2+a_1xy+a_3y=x^3+a_2x^2+a_4x+a_6$$ Why are the coefficients named $a_1, a_2, a_3, a_4$ and $a_6$ in this manner, corresponding to $xy, x^2, ...
2
votes
1answer
90 views

Number of points on $Y^2 = X^3 + A$ over $\mathbb{F}_p$

Let $p\equiv 2\pmod{3}$ be prime and let $A\in\mathbb{F}^{∗}_p$ . Show that the number of points (including the point at infinity) on the curve $Y^2 = X^ 3 + A$ over $\mathbb{F}_ p$ is exactly $p + 1$ ...
2
votes
1answer
248 views

Modular functions and elliptic functions

Does anybody know of an equation formally equating modular functions and elliptic functions similar to Euler's equation for exponential and trigonometric functions? Any advice much appreciated. ...
1
vote
0answers
48 views

Is it possible to say that every point $P$ in $C(ℚ)$ other than the 'basis' is of finite order?

Let $C$ an elliptic curve over $\mathbb Q$. Assume that the rank of $C(ℚ)$ is equal to $r$. Then the cardinality of a maximal independent set in $C(ℚ)$ is $r$, thus there exists $r$ independent points ...
1
vote
1answer
81 views

Violating assertion in Cohen's instructions for Weierstrass reduction

I am trying to follow case 2 of the procedure given in Cohen: for the cubic $f(x,y,z) = x^3 + 3 y^3 - 11 z^3$ using the rational point $P_0 = (2 : 1 : 1)$. The tangent at this point is $y = - ...
0
votes
0answers
59 views

Can we extend the map $φ$ to $ℝ^{r}×C(ℚ)^{\text{tors}}→C(ℚ)$ as an isomorphism or not?

The motivation to this question can be found in How I can express $(x,y)∈G$ by using the $r$ independent points $P_1,P_2,\ldots,P_r$ We know that there is an isomorphism ...
35
votes
3answers
624 views

The resemblance between Mordell's theorem and Dirichlet's unit theorem

The first one states that if $E/\mathbf Q$ is an elliptic curve, then $E(\mathbf Q)$ is a finitely generated abelian group. If $K/\mathbf Q$ is a number field, Dirichlet's theorem says (among other ...
23
votes
3answers
678 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 ...
12
votes
2answers
587 views

How are the Tate-Shafarevich group and class group supposed to be cognates?

How can one consider the Tate-Shafarevich group and class group of a field to be analogues? I have heard many authors and even many expository papers saying so, class group as far as I know is ...
8
votes
1answer
2k views

Reading the mind of Prof. John Coates (motive behind his statement)

To start with the issue, I have been thinking from many days that Birch-Swinnerton-dyer conjectures should have some association with the Galois theory, but one day I got the Article of Tate called as ...
3
votes
1answer
72 views

rational points on particular elliptic curve

I do have a few books that discuss elliptic curves, however... What are the rational points on $$ y^2 = 4 x^3 - 4 x = 4 x(x-1)(x+1)? $$ I think it ought to be $(-1,0), (0,0), (1,0).$ Maybe it's ...
10
votes
3answers
344 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 $x^2+2=y^3$. ...
9
votes
2answers
214 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 ...
8
votes
2answers
470 views

Is the real locus of an elliptic curve the intersection of a torus with a plane?

In Lawrence Washington's book Elliptic Curves: Number Theory and Criptography I read that if $E$ is an elliptic curve defined over the real numbers $\mathbb{R}$ then the set of real points ...
7
votes
2answers
482 views

Geometric reason why elliptic curve group law is associative

The question title says it all. I am looking for a geometric proof for the fact that the group law defined on elliptic curves is associative. I've heard somewhere about something on the ...
5
votes
1answer
625 views

How do I show that this curve has a nonsingular model of genus 1?

Let $C$ be the projective closure of $Z(f) \subset \mathbf{A}^2$ where $f$ is an irreducible polynomial of degree 4 in $x$ and degree 2 in $y$, so $C = Z(f^*) \subset \mathbf{P}^2$ where $f^*$ is the ...
4
votes
2answers
118 views

There is no Pythagorean triple in which the hypotenuse and one leg are the legs of another Pythagorean triple.

According to Wikipedia, There are no Pythagorean triples in which the hypotenuse and one leg are the legs of another Pythagorean triple. I cannot find the proof in the citation provided. I am ...
4
votes
2answers
975 views

What is a primitive point on an elliptic curve?

While working with elliptic curves for cryptography reasons, I found the notion of a primitive point, but no definition. For example, $P(0,6)$ is a primitive point on the elliptic curve $y^2\equiv ...
3
votes
3answers
180 views

Reference: Elliptic curves as complex tori

I'm looking for books which contain a more or less self-contained description of how elliptic curves over $\mathbb{C}$ - that is, nonsingular plane cubic curves - can be realized as a quotient of the ...
5
votes
4answers
295 views

upper bound on rank of elliptic curve $y^{2} =x^{3} + Ax^{2} +Bx$

I was told the following "Theorem": Let $y^{2} =x^{3} + Ax^{2} +Bx$ be a nonsingular cubic curve with $A,B \in \mathbb{Z}$. Then the rank $r$ of this curve satisfies $r \leq \nu (A^{2} -4B) +\nu(B) ...
4
votes
3answers
211 views

The rational points on the curve: $y^2=ax^4+bx^2+c$.

I wonder how to find the rational points on the curve: $y^2=ax^4+bx^2+c$. Is there infinite rational points on this curve? For example:$y^2=x^4+3x^2+1.$If we set $y=x^2+k$,then $2kx^2+k^2=3x^2+1$, ...
2
votes
2answers
164 views

How to prove $~\sqrt{3}\sqrt{4a^{3}-1}~$ isn't an integer?

I'm trying solve: $~a^3 + b^3 = c^3~$ has no nonzero integer solutions. If $~(c−b)=1~$ then $~c^3-b^3=3c^2-3c+1=a^3,~$ from Wolframalpha get: $$ c = \dfrac{3- \sqrt{3}\sqrt{4a^{3}-1}}{6} \\ c = ...
2
votes
1answer
138 views

Proving the condition for two elliptic curves given in Weierstrass form to be isomorphic

I'm taking a course on elliptic curves and trying to understand the proof of Proposition 3.2. Let $E$, $E'$ be elliptic curves over $K$ in Weierstrass form: ...
1
vote
2answers
146 views

$N^2=2M^4-2p^2e^4$ has no integer solution

If $\gcd(M,e)=\gcd(N,e)=1$ and $p$ is prime and $p‎\equiv 5 \mod(16)$ then how I can show that $N^2=2M^4-2p^2e^4$ has no integer solution.
0
votes
0answers
23 views

How to find the tangent vector to a curve at a special point

The two points $A=(x_{0},y_{0},z_{0})$ and $B=(x_{1},y_{1},z_{1})$ are given. The line segments $AC$ and $BC$ make equal angle $\alpha$ with the horizontal plane through $C$. The angle ...
0
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0answers
44 views

How to find the subgroups of the group $C(ℚ)$?

Let $C$ be a fixed elliptic curve over $ℚ$. The group $C(ℚ)$ is a finitely generated Abelian group and we have $$C(ℚ)≃ℤ^{r}⊕C(ℚ)^\mathrm{tors}$$ where $C(ℚ)^\mathrm{tors}$ is a finite abelian group ...
6
votes
1answer
135 views

explict form of the equation of elliptic curve

Let $E(\mathbb{F}_{q^2})$ is elliptic curve with #$E(\mathbb{F}_{q^2}) =q^2 + q + 1$. Can we write equation of this curve in the explicit form?
5
votes
0answers
127 views

Attacking Elliptic Curve Cryptography Problem with a Bad Reduction $\pmod p$

I'm working on a crypto problem as a puzzle and unfortunately my math isn't at the level I need it to be to answer the question. I have been given a prime $p$, a curve $E$ defined over $F(p)$, a ...
3
votes
1answer
232 views

The Process of Choosing Projective Axes to Put an Elliptic Curve into Weierstrass Normal Form

I'm reading the book "Rational Points on Elliptic Curves" and on page 23 the author takes an arbitrary (non-singular) elliptic curve in the projective plane and finds a rational point $O$, referring ...
3
votes
2answers
3k views

How elliptic arc can be represented by cubic Bézier curve?

If I have an arc (which comes as part of an ellipse), can I represent it (or at least closely approximate) by cubic Bézier curve? And if yes, how can I calculate control points for that Bézier curve?
2
votes
1answer
64 views

History of the Coefficients of Elliptic Curves — Why $a_6$? [duplicate]

I would like to know what is the motivation behind the naming convention of the Weierstrass form of elliptic curves given as $$E:y^2+a_1xy+a_3y=x^3+a_2x^2+a_4x+a_6.$$ I can see that $a_1,a_2,a_3,a_4$ ...
2
votes
2answers
240 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 $\Bbb F_{11}$ ...
2
votes
1answer
200 views

writing down the minimal discriminant of an elliptic curve

Let $j$ be an integer. Does there exist an elliptic curve $E_j$ over $\mathbf{Q}$ with $j$-invariant equal to $j$ whose minimal discriminant we can write down in a practical way? For example, can ...
2
votes
1answer
261 views

Form of minimal integral Weierstrass equation for elliptic curve over $Q$ with good reduction at $2$ and $3$.

If $E$ is an elliptic curve over $\mathbb{Q}$ which has good reduction at $2$ and $3$, is it always possible to find a minimal integral Weierstrass equation for $E$ of the form $y^2 = x^3 + Ax + B$ ...
2
votes
2answers
231 views

Trying to piece together an integral addition theorem

If we have a curve $C:\{ P(x,y) = 0 \}$ and define $\omega=\frac{\mathrm{d}x}{y}$ then is $$\int_0^A \omega + \int_0^B \omega = \int_0^{A \oplus B} \omega$$ (with $\oplus$ being addition on a group ...
1
vote
1answer
477 views

Doubling a point on an elliptic curve

I've a programming background and am just about to get into a project where Elliptic Curve Cryptography (ECC) is used. Although our libraries deal with the details I still like to do background ...
1
vote
0answers
65 views

Frobenius endomorphism on supersingular elliptic curve

Let we have supersingular curve $E(\bar{\mathbb{F}_q})$. Is it true that for every point $P$ $q$-Frobenius endomorphism $\pi_q$ can be write as $A + [q]B$ where $P = A + B$? It is true if ...
1
vote
1answer
175 views

The Group of points on the Elliptic curve $y^2=x^3+1$ over $\mathbb{F}_5$

So I'm trying to understand the group of points of $y^2=x^3+1$ over $\mathbb{F}_5$ and for some reason I seem to be getting nonsense answers and I'm not sure what I'm doing wrong. So basically my ...