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

learn more… | top users | synonyms

3
votes
0answers
45 views

Complex multiplication and ray class fields

This question is mainly referring to the proof of Theorem 5.6, Chapter 2 of Silverman's "Advanced Topics in the AEC". Basically, let $K$ be an imaginary quadratic field, and $E$ be an elliptic curve ...
1
vote
1answer
26 views

Confusion about holomorphic differential on elliptic curve

Let $(a:b)\in\mathbb{C}P^1$ and look at the elliptic curve $C$ given by $y^2=x^3+a^4x+b^6$. It is well known that on this elliptic curve we have the holomorphic differential $dx/y$. I have two ...
4
votes
0answers
41 views

How to compute explicitly the covering map in the modularity theorem?

The modularity theorem (original Shimura-Taniyama-Weil conjecture) asserts the existence of a covering (uniformization) map $\pi:X_0(N) \to E$ for every $E$, an elliptic curve defined over ...
0
votes
0answers
39 views

Show $(x,y) \rightarrow (x,-y)$ is a group homomorphism?

Show that $(x,y) \rightarrow (x,-y)$ is a group homomorphism from $E$ to itself where $E$ is an elliptic curve in Weierstrass form. So $E$ is of the form $y^2=x^3+ax+b$. Would I just show that ...
3
votes
1answer
65 views

Reference request: Fibre functor for elliptic curves is pro-representable

I am writing a project on étale fundamental groups of elliptic curves and I want to include a proof of a key theorem: the fibre functor on the category of finite étale covers of an elliptic curve is ...
2
votes
1answer
26 views

Computing the multiplicative inverse for point addition on an elliptic curve

I'm trying to perform point addition on an elliptic for two points taken from an example in the book "Understanding Cryptography by Christof Paar & Jan Pelzl". The points I'm trying to add are: ...
2
votes
0answers
71 views

Quartic Diophantine equation $ 2 x^4 - 2 x^2 = 3 (y^2 - 1)$

About the quartic Diophantine equation: $$ 2 x^4 - 2 x^2 = 3 (y^2 - 1)$$ On oeis.org/A180445 it says that all positive solutions $(x,y)$ are: $$(1,1)\ \ (2,3)\ \ (3,7) \ \ (6,29)\ \ (91,6761)$$ ...
0
votes
2answers
41 views

Isogenous elliptic curves over finite fields have the same number of points

I'm stuck in this question, it is the first part of exercise 5.4 from Silverman - The arithmetic of elliptic curves. Let $C,D$ be two isogenous elliptic curves over a finite field $\mathbb{F}_q$. ...
0
votes
0answers
17 views

Translating from (short) Weierstrass form to a Lattice. [duplicate]

I know that given an elliptic curve in the form $\mathbb{C}/L$ for L some lattice, $\mathbb{Z}+\mathbb{Z}\tau$, then we can use the Weierstrass $\wp$ functions to turn it into short Weierstrass form, ...
1
vote
1answer
52 views

What does extra zero of an $L$-function mean?

This is a very vague question. What does an extra zero of an $L$-function mean? There are lots of papers written on this topic, investigating the extra/exceptional zeros of various $p$-adic ...
2
votes
0answers
38 views

Degree $5$ embeddings of genus 1 curves, plucker embedding of a Grassmannian…

Here is something Ravi Vakil says after 19.9F in his notes. He is talking about embeddings of a genus 1 curve $C$ by complete linear series associated to the divisors $O(nP)$. "The beautiful ...
3
votes
1answer
68 views

Is a projective curve minus finite number of points affine?

The answer to the question in the title is "yes". This is proved for example here, or here (page 5). However, I seem to be able to construct a counterexample. Can you help me find the flaw in my ...
0
votes
0answers
50 views

question about Galois theory and dimension

I try to understand the proof of the lemma 13.7 of the following article: http://www.cs.nyu.edu/courses/spring05/G22.3220-001/ec-intro1.pdf The lemma says that if $r$ is a rational function which ...
0
votes
0answers
30 views

Points of finite order : Size of the group E/E0

My curve is given by E : $y^2 = x^3-3267x+45630$. Bad primes are 2,3,17. I want to find the size of group $E/E_0$. I know that $E_0(Q_2)$ are points on $E(Q_2)$ that do not reduce to a singular point. ...
2
votes
1answer
48 views

Order of the pole of projective curve using uniformizer

I need to find divisor of functon $f=y$ i.e. $\frac{y}{z}$ for projective curve $y^2z=x^3-xz^2$ and I have some questions: For example, I have pole: $(0:1:0)$ and zeros $(0:0:1),(1:0:-i),(1:0:1)$. It ...
1
vote
0answers
57 views

Plotting Elliptic Curve over Finite Field in Maple

Not sure if I might be in the wrong section, but I am looking for guidance on how to plot an elliptic curve over a finite field in Maple. I have tried looking it up but only getting good results for ...
0
votes
1answer
36 views

Hasse theorem proof

I don't understand the following points 1) why is $E$ isomorphic to the $\ker(\phi - 1)$? 2) Why is $\#\ker(\phi - 1) = \deg(\phi - 1)$? The proof is taken from here page 11 ...
0
votes
0answers
36 views

ELMO 2012 Shortlist N9

I'll admit that I've made no progress to solve this one. It is way too hard. I guess I must do some stuffs with elliptic curve to solve it but I got nowhere So, here is the problem: Are there ...
0
votes
0answers
20 views

degree of isogeny understanding

In the context below why is the degree of the Frobenius endomorphism p ?
2
votes
1answer
48 views

$p$-depletion of a modular form

Let $p$ a prime and $N$ an integer such that $p\not\mid N$. I will denote with $X_0(m)$ the modular curve with respect to the congruence subgroup $\Gamma_0(m)$. Let $f$ be a modular form with ...
0
votes
0answers
25 views

Mordell-Weil group - 2 descent

On my elliptic curve, I have generator P and 2-torsion point T in general. If i compute points nP and nP+T, and substitute these points in my function,z, I noticed that the non-square part are ...
2
votes
2answers
57 views

8 divides $\#C(\mathbb{F}_p)$ for $C$ : $y^2=x(x+1)(x-8)$, $p\geq 5$

I was trying to compute the torsion of $C$ : $y^2=x(x+1)(x-8)$ over $\mathbb{Q}$ by using the fact that the order of the torsion divides the order of $C$ reduced modulo any $p\nmid 2\Delta$. For any ...
2
votes
1answer
211 views

Looking for help with this elementary method of finding integer solutions on an elliptic curve.

In the post Finding all solutions to $y^3 = x^2 + x + 1$ with $x,y$ integers larger than $1$, the single positive integer solution $(x,y)=(18,7)$ is found using algebraic integers. In one of the ...
1
vote
2answers
40 views

Is $H^{1}(k,E[n])$ a subgroup of $H^{1}(k,E)$?

Let $E/k$ be an elliptic curve. Consider $E[n]$ which is a subgroup of $E$. Is it true that $H^{1}(k,E[n])$ is again a subgroup of $H^{1}(k,E)$ in Galois cohomology? I thought that this was true but ...
3
votes
1answer
69 views

Are there any other integer points on the elliptic curve $Y^2 = X^3 + 1$ beyond $(-1, 0), (0, \pm 1), (2, \pm 3)$?

The charm of elliptic curves is that given one or two integer points, one can find others by the group law. However the easy to guess points from the title just pump me around trough a cyclic group of ...
1
vote
1answer
31 views

Does adjoining a $p$-power divisor to an elliptic curve, where $p\neq 2$ is a prime, always result in a Galois group of order greater than $2$?

The question says it: Suppose I have a field $K$, whose characteristic is, for simplicity, zero, and an elliptic curve $E$ over $K$, and $x\in E(K)$. Suppose that $p$ is a prime different from $2$, ...
4
votes
1answer
145 views

Why does Lenstra ECM work?

I came across Lenstra ECM algorithm and I wonder why it works. Please refer for simplicity to Wikipedia section Why does the algorithm work I NOT a math expert but I understood first part well enough ...
3
votes
0answers
80 views

Kummer map and cohomology group for an elliptic curve

Let $E=E_q$ be the Tate ellipitc curve over a finite extension $K$ of $\mathbb{Q}_p$ for a $q$. Let $T$ be its p-adic Tate module. Let $\mathfrak m$ be the maximal ideal in $K$. I saw in this paper ...
3
votes
1answer
46 views

Finding quadratic twist of elliptic curve

Given a elliptic curve over $F_p$ with the equation $E : y^2 = x^3 + Ax + B$, I want to find an isomorphous curve (quadratic twist) which can be written in the form $E': y^2 = x^3 + A'x + B'$ where ...
0
votes
0answers
10 views

How to store tables for ECM stage 2

This question about realization ECM stage 2 on GPU. I now that there exists some optimization for the stage 2 of ECM. Namely, let $N$ be a composite number, $q|N$ be a prime, $P=(x_P::z_P)$ be a point ...
3
votes
0answers
57 views

Automorphism of elliptic curve, Vakil 19.10.E

There are many other proofs of finding all the possible automorphism groups of elliptic curves, but I am interested in the $Hint$ and the corresponding proof in the following exercise from Vakil's ...
1
vote
0answers
29 views

Elliptic curve Schoof algorithm, projective polynomial point coordinates

I'm trying to understand Schoofs algorithm for determining $\#E(F_P)$ of an Elliptic curve $y^2 = x^3 + ax + b$ over $F_P$. For this I'm looking at the implementation of MIRACL: ...
2
votes
1answer
108 views

Generalizing Fermat's challenge to Frenicle

In 1643, Fermat asked Frenicle et al to find a special Pythagorean triple $a,b,c$ such that for $n=1$, $$a+nb = r_1^2\\ a^2+b^2 = r_2^4\tag1$$ Equivalently, ...
12
votes
1answer
190 views

For which integers $a,b$ does $ab-1$ divide $a^3+1$?

A problem I wasn't able to solve: For which values of $a,b\in\mathbb{Z}$ does $ab-1$ divide $a^3+1$? I am looking for every possible solution. Some of them are trivial, like $a=0,b=0$ or ...
1
vote
1answer
40 views

Trace of a function on an elliptic curve

Let $K/F$ be a Galois extension of number fields with Galois group $G$. Let $E$ be an elliptic curve defined over $F$ and $f \in K(E)^{\times}$ be a function. Define the trace of $f$ to be ...
1
vote
0answers
57 views

Weak Mordell-Weil over number fields

I have a question regarding the Mordell Weil theorem a number field $K$. I read the proof of the Mordell Weil theorem in "rational points on elliptic curves" by Tate and Silverman. They presented a ...
3
votes
1answer
100 views

Special Euler bricks and $x^2(y^2-1)^2+y^2(x^2-1)^2=z^2$

Define, $$P_1 := a^2+b^2\\ P_2 := a^2+c^2\\ P_3 := b^2+c^2$$ Let, $$a,b,c = 2xy,\;x(y^2-1),\;y(x^2-1)$$ and $P_1,P_2$ become squares. If we wish to make $P_3$ a square as well, then, ...
0
votes
1answer
44 views

Genus of $3x^{3}+4y^{3}+5z^{3}$

I am supposed to calculate the genus of the projective curve $3x^{3}+4y^{3}+5z^{3}$, then use Weil conjecture. I don't know how to calculate the genus, it seems like plugging in formula just give me ...
1
vote
0answers
71 views

Pictures of curves over finite fields with many points

At the manypoint page for $2^3$, genus=3, there is the note: "In his Harvard notes, Serre notes that a model of the Klein curve gives an example of a genus-3 curve with 24 points over $F_8$: $(x + y ...
0
votes
0answers
18 views

Weiertrass equation of an elliptic curve.

We know that every elliptic curve is a non-singular $\textbf{cubic}$ projective curve (curve of genus 1), but we can transform this in the Weiertrass form $$y^2 + a_1xy + a_3y = x^3 + a_2x^2 +a_4x ...
1
vote
0answers
56 views

Etale map from a variety to an elliptic curve

I read this sentence and I can't see why it is true. Let $E$ be an elliptic curve over an algebraically closed field $k$, $f\colon Y\to E$ an etale map; then $Y$ is also a curve over $k$. Can ...
3
votes
2answers
54 views

What happens in elliptic curve primality testing if you cannot find a suitable discriminant?

I'm trying to understand the computational aspect of elliptic curve primality testing (specifically the Atkin-Morain test), and in general, I understand why it works for a prime number. However, when ...
2
votes
1answer
60 views

How to find the equation of the curve defining the intersection of two quadrics.

Let $\mathbb{F}_{q}$ be the finite field of order $q$ where $q$ is the power of an odd prime. Let $u$ be a fixed non-square in $\mathbb{F}_{q}$ and let $\lambda\in\mathbb{F}_{q}$ such that ...
2
votes
1answer
62 views

Discriminant of Elliptic Curves

In the study of elliptic curves, specifically in Weierstrass form, you have the equation $E : y^2 = x^3 +ax +b$. However I have found the discriminant comes in two different forms: $\Delta = ...
4
votes
1answer
94 views

What are numbers $n$ such that $a^2+nb^2 = c^2$ and $na^2+b^2 = d^2$?

Let $n$ and $a,b,c,d,$ be in the positive integers. I. For the system, $$a^2-nb^2 = c^2\\a^2+nb^2=d^2$$ then $n$ is a congruent number. The sequence starts as $n=5,6,7,13,14,15,20,21,$ and so ...
1
vote
1answer
37 views

Discriminant of an elliptic curve

I have found different discriminants for general Weierstrass elliptic curves: $y^2 = x^3 + ax + b$ For example, WolframAlpha state it is $-16(4a^3 + 27b^2)$ on their site ...
1
vote
0answers
26 views

How to find points at infinity in projective coordinates for the $x^2 -3xy-2y^2-x+1=0$

How to find points at infinity in projective coordinates for this algebraic curve $x^2 -3xy-2y^2-x+1=0$ I don't know how to start! Any help will be appreciated.
1
vote
1answer
26 views

How to prove that $f(x)$ has a multiple root in $Q$ if and only if $disc(f) =0$

Let $f(x) = x^3 +ax+b$ contained in $Q[x]$ prove that $f(x)$ has a multiple root in $Q$ if and only if $disc(f) =0$ This is what I've so far since $f(x) = (x-A_1)(x-A_2)(x-A_3), A_1,A_2, A_3$ are ...