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

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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
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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
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1answer
27 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
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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: $$(1,1)\ \ (2,3)\ \ (3,7) \ \ (6,29)\ \ (91,6761)$$ ...
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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, ...
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2answers
43 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 $\mathbb{F}_q$. ...
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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 ...
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1answer
56 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 ...
3
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1answer
71 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 ...
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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 ...
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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. ...
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1answer
49 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 ...
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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 ...
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38 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 ...
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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 ...
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20 views

degree of isogeny understanding

In the context below why is the degree of the Frobenius endomorphism p ?
2
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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 ...
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26 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
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2answers
58 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 ...
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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 ...
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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 $2$, ...
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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 ...
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1answer
47 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 ...
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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
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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 ...
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1answer
114 views
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35 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: ...
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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, ...
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58 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 ...
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1answer
45 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 ...
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1answer
198 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 ...
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1answer
41 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 ...
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73 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 ...
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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 ...
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1answer
101 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, ...
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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 ...
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2answers
55 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 ...
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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 ...
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1answer
61 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 ...
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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.
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1answer
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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 ...
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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 ...
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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 = ...
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1answer
29 views

Isomorphism on cubics group law

Let $C$ a non singular cubic projective plane curve with a fix point $O$, we use que chord-tangent method to make $G = (C,+,O)$ an abelian group, if I choose another fix point $O'$ and construct the ...
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1answer
50 views

Order of a point on elliptic curve

I am trying to prove that the following elliptic curve has rank=1: $$y^2=x^3+x^2+x+1$$ From the map $$\delta: E(Q)\rightarrow Q(i)^*/(Q(i)^*)^2$$ $$(x,y)\mapsto (x-i) $$ for $(x,y)\neq O$ and ...
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3answers
59 views

Suggestions for readings; Elliptic curves over function fields

I would love to know some good refercences about Elliptic curves over function fields. Especially in view with Mordell-Weil's Theorem. I am already familiar with the main proof of Mordell's theorem in ...
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0answers
23 views

Complex Atlas for Elliptic Curves over $\mathbb{C}$

I know that every elliptic curve over $\mathbb{C}$ is isomophic to a torus $\mathbb{C}/\Lambda$ in the sense of Riemann Surfaces, moreover $E(\Lambda)$ as topological subspace of ...
2
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1answer
67 views

Divisor on curve of genus 2

Let $C$ be a smooth, projective curve of genus 2. I want to show that there exists a non-constant rational function $f \in k(C)$ having divisor of the form $$(f) = P_1 + P_2 - P_3 - P_4 $$for points ...
2
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1answer
33 views

Computing the dual of frobenius endomorphism

There is an exercise in my course Elliptic Curves and I am not sure if I am doing it right. The question is as follows: Let $E$ be the elliptic curve over $\mathbb{F}_2$ given by $Y^2+Y=X^3$. (a) ...
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26 views

What is the right dual isogeny?

I have a question regarding dual isogenies. I read an example in Silverman's book about elliptic curves and am wondering something about this example. We have $\zeta$ as a primitive cube root of ...