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

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Galois invariants of the Tate module of an elliptic curve over a number field

Let $K$ be a number field, $E$ be an elliptic curve over $K$, $l \neq p$ be two different prime numbers and $v$ be a place of $K$ above $l$. I am trying to understand the proof of proposition I.6.7 ...
2
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0answers
53 views

Isogenies between elliptic curves with specified torsion groups

For each of the $15$ possible torsion groups of an elliptic curve defined over $\mathbb{Q}$ we have an infinite family of curves with that torsion group. This sometimes goes under the name of Kubert ...
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0answers
19 views

Order's size (in bits) of an elliptic curve

I am trying to prove that, given an Elliptic Curve defined on $\mathbb{F}_p$ with $p$ a prime number, the order $q$ verifies: $|p| \le |q| \leq |p|+1$ where $|x|$ denotes the length in bits of $x$. ...
2
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1answer
24 views

Polynomial representation of elliptic curve points (Frobenius Endomorphism)

I'm trying to understand the Schoof algorithm for counting the number of points on elliptic curves in finite fields. I.e. the most basic algorithm to efficiently determine $\#E(F_p)$. For literature, ...
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0answers
44 views

Elliptic curve is self dual.

How to prove $E[p^\infty] \cong Hom ( T_E, \mathbb{Q}_p/\mathbb{Z}_p(1)) $ where $T_E$ denotes the Tate module of $E$ ?
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21 views

Optimal bounds of given curve [closed]

How to find optimal bounds for the heights of integral solutions on a given any Weierstrass equation? Please explain with an example. regards, S. Nair.
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28 views

How to find the affine coordinate algebra of $n$-torsion points of elliptic curve?

In particular, I'm wondering about the affine coordinate algebra of $E[3]$ where $E$ is the elliptic curve $y^{2}=x^{3}-D$ over $\mathbb{Q}$ with $D=2^{8}3^{5}5^{2}$. I think we can view $E[3]$ as ...
1
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1answer
39 views

Proving $x^2 - x = y^5 - y$ is a hyperelliptic curve

Greetings to one an all! How can we prove the curve "$x^2 -x = y^5-y$" is a hyperelliptic curve? Is a hyperelliptic curve the same as a hyperbolic elliptic curve or are there any differences?
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The form of the zeta function of an elliptic curve over a finite field

I seek a (very) elementary proof that the zeta function of an elliptic curve $E$ over $\mathbb{F}_q$ has the form $$Z(T)=\frac{1-aT+qT^2}{(1-T)(1-qT)}.$$ Something tedious and computational making use ...
0
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1answer
16 views

Addition of x-coordinate on elliptic curve given by Möbius Transformation

Consider the elliptic curve $y^2=(x-\alpha)(x^2+ax+b)=x^3+(a-\alpha)x^2+(b-a\alpha)x-\alpha b$ over the field $K$ with $\text{char}\ K\not= 2$. The questions I am doing asks for a formula for the ...
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2answers
71 views

The cardinality of elliptic curves over finite field

Given an elliptic curve over $\mathbb Q$ as $y^2=f(x)$ where $f(x)$ is a cubic polynomial. In some places I read that if $p$ is a prime of good reduction then we have that $E(\mathbb F_p)=p+1$. Is ...
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43 views

Algebraic independence of `Riemann-Roch' elements

First of all, I'm not too sure on what terminology should be used in the title: the question deals with the vector spaces $$\mathcal{L(D)}=\{f\colon E\to\mathbb{C} \mid f\text{ is meromorphic}, ...
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0answers
93 views

Is it normal surface of general type to have infinitely many positive rank elliptic curves?

I am not good at algebraic geometry and almost surely am misunderstanding something. Got an alleged argument against Bombieri-Lang conjecture and would like to know what the mistake is. One of the ...
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2answers
23 views

Problem with Elliptic Curve in Montgomery form

I am trying to understand how points are added in Elliptic Curves in Montgomery form. I am working with the curve $$3y^2 = x^3 + 5x^2 + x \mod 65537$$ Adding the point $(3,5)$ with itself gives (or ...
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0answers
28 views

Why do we assume the ring to be torsion free when dealing with formal logarithms in the context of formal group laws?

Let $F$ be a formal group over a ring $R$. Why do we require that $R$ has no additive torsion before we discuss formal logarithms?
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1answer
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Calculating Elliptic Curve cofactor h

An Elliptic Curve in short Weierstrass form over a finite field $F_p$ is given by the equation: $$y^2 = x^3 + ax + b \mod p$$ To use this curve for cryptographic purposes, in the domain parameters ...
3
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1answer
31 views

Galois Representation with $D_{10}$ image

I want to construct an explicit elliptic curve $E$ over a number field $K$ such that $Gal(K(E[l])/K) \cong D_{10}$ where $D_{10}$ is the dihedral group of order 10 and $l$ is a prime number. ...
2
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1answer
33 views

height of formal group of an elliptic curves

I have an elliptic curve $E$ defined over a complete discrete-valued field $K$ of characteristc $0$. the residue field $k$ is of positive characteristic $p$. Then $E[p]=\mathbb{Z}/p\mathbb{Z} \times ...
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1answer
43 views

Show that Weierstrass function is elliptic function.

Prove that Weierstrass function is periodic with respect to lattice $L (L\subset \mathbb{C})$ .i-e $f(z+w,L)=f(z,L)$ ($w\in L$). $f(z,L)=\frac{1}{z^2}+\sum_{0\ne w\in ...
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1answer
25 views

Point conversion between Twisted Edwards and Montgomery curves

With the great help of Birational Equvalence of Twisted Edwards and Montgomery curves I know how to convert twisted Edwards curves into their birationally equivalent Montgomery counterparts where I ...
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1answer
132 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 ...
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1answer
28 views

can someone explain Nagell-Lutz theorem

(elliptic curve $y^2 = x^3 + ax^2 + bx + c$) Nagell-Lutz theorem: If $p(x, y)$ is finite order on a given integer coefficient elliptic curve satisfy: (1) x and y are integer (2) y = 0 or y | D (D ...
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1answer
28 views

Birational Equvalence of Twisted Edwards and Montgomery curves

I'm trying to understand the birational equivalence between Twisted Edwards and Montgomery curves and try to calculate some examples. In particular, as an example, I'm looking at the Ed25519 Twisted ...
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2answers
54 views

Reduction map on torsion points of an elliptic curve and their valuation

Let $K$ be a field of characteristic zero, complete with respect a discrete valuation $v$. Assume that the residue field $k$ is of positive characteristic $p$. Now take an elliptic curve $E$ defined ...
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Structure of first-coordinate-projection of set of solutions of “elliptic” diophantine equation $xy(6-(x+y))=6$

Say that a rational number $a$ is good iff there is a rational number $b$ with $ab(6-a-b)=6$, or equivalently iff $a^4 - 12a^3 + 36a^2 - 24a$ is the square of a rational number. Denote by $G$ the set ...
3
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1answer
36 views

Group operations on Montgomery Curves in affine representation

I'm trying to understand group operations on elliptic Montgomery curves in affine representation. Let's say the curve I use is Curve25519, i.e.: $$y^2 = x^3 + A\,x^2 + x\quad\text{where}\quad ...
5
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2answers
124 views

Homogeneous diophantine equation $x^3+2y^3+6xyz=3z^3$

Is it known if there are infinitely (non-proportional) many integer solutions to $x^3+2y^3+6xyz=3z^3$ ? Motivation : if true, this would provide an alternative solution to that recent MSE question, ...
3
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1answer
79 views

Why do the Diophantine equation $\frac{x}{y}+\frac{y}{z}+\frac{z}{x}=n$ gives an elliptic curve?

In a book "Which way did the bicycle go" was tought a problem of integer solutions of certain Diophantine equation. This is the idea, not an exact quotation: For which integers $n$ are there integers ...
3
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1answer
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Definition of Selmer-Group for Elliptic Curves

Im facing a problem in Silvermans Book "Arithmetic of elliptic Curves" at the beginning of chapter X.4 concerning the exact sequences. Let $K$ be a number field with a valutaion $v$. I'm ...
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26 views

Arithmetic data in an elliptic curve over a field $\mathbb K$

Note: In this context, $E(K)$ denotes an elliptic curve $E$ over a number field $K$, and $L(E,s)$ denotes the Hasse-Weil $L$ function. Is the rank of the abelian group $E(K)$ of points of $E$ the ...
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What is the group structure on the ring of power series around a point that makes it “the completion of an elliptic curve” along that point?

I've been struggling to understand the explicit details of the completion of an elliptic curve about the origin, and am desperately confused by the explicit details of the resulting group operation. ...
3
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32 views

Weierstrass-$\wp$ Function Asymptotics

Given the Weierstrass-$\wp$ function, $$\wp(2x+1+\tau \mid 1, \tau),$$ with half-periods $1$ and $\tau=\omega_2/ \omega_1$, I want to look at the case where $\rm{Re}(\tau) \in \mathbb{Z}$ and I want ...
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231 views

rationality of $\ell$-adic representation attached to an elliptic curves

Let $E$ be an elliptic curves defined over a number field $K$. Consider the $\ell$-adic representation attached to $E$ $$ \rho_{\ell}:\mathrm{Gal}(\overline{K}/K) \longrightarrow ...
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1answer
80 views

Solve for $x$ in elliptic curve $y^2 = x^3 + ax + b$

Given $y$, is it possible to solve for $x$ in the elliptic curve equation $y^2 = x^3 + ax + b$ over a finite field? Or is it known to be as difficult as say, something like the discrete logarithm ...
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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 ...
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2answers
50 views

Third point of intersection is also a point of inflection?

Let $C \subset \mathbb{P}_2$ be a nonsingular cubic. If $L$ is a line through two distinct points of inflection on $C$, how do I show that the third point of intersection is also a point of ...
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1answer
58 views

Is this an elliptic curve?

I am trying to learn what elliptic curves are. Sofar I have not had any luck understanding when a curve is elliptic and when it is not. Is this an elliptic curve? $$y^2 = \frac{x}{4}-\frac{17 ...
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Remark 4.23.4 in Hartshorne.

Remark 4.23.4 in Hartshorne references a paper by Elkies that explains that$$\mathfrak{B} = \{p \text{ prime}: X_{(p)} \text{ is nonsingular over }k_{(p)}, \text{ and }X_{(p)}\text{ has Hasse ...
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42 views

Representations of algebraic group ($S_{\mathfrak{m}}$)

I'm studying Serre's book "Abelian $\ell$-adic Representations and Elliptic Curves" and in chapter II $\S$2.4 we have this proposition: Consider $v$ a finite place of $K$ and $F_v \in Gal(K^{ab}/K)$ ...
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1answer
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Galois conjugation in $\mathbb{Z}/m\mathbb{Z}$

In Silverman's Arithmetic of Elliptic Curves, he introduces the Weil pairing as a means of making the determinant pairing Galois invariant. He writes that $\det(P^{\sigma},Q^{\sigma})$ and ...
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1answer
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How do you prove that rational points on $y^2 = x^3 - 2$ are of the form $(A/B^2, C/B^3)$, where are $A, B, C$ are coprime?

I was only browsing this book on number theory and the author shows how the solution $(3, 5)$ can be used to generate other exotic rational solutions and then in the end leaves the problem I'm asking ...
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1answer
54 views

When the trace of the Frobenius homomorphism is zero?

Let's consider an elliptic curve over a finite field $\mathbb F_p$. The trace of the Frobenius homomorphism is defined as: $$a_p=p+1-\#E(\mathbb F_p)$$ See for example here. I read that this value ...
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Framing a lattice problem from information available on multiple runs of GLV decomposition

I have posted a similar question here. The GLV method [ref] is used to speed up ECDSA signature generation. In this method, an input scalar $k$ is decomposed into two scalars, $k_1$ and $k_2$. Then ...
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1answer
36 views

Integer points belonging to two distinct elliptic curves.

Two different circles can have an integer point in common (for example, $P=(1,1)$ belongs to both $x^2+y^2-2=0$ and $x^2+y^2-4(x+y)+6=0$) but any pair of distinct elliptic curves on the class defined ...
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1answer
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Animation of Weierstrass $\wp$-function as a map from a torus to the sphere?

I am wondering if there exists somewhere an "animation" of one such map (for some lattice / torus), in the style of the kind of $z \mapsto z^2$ maps one encounters in complex analysis classes (one can ...
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1answer
24 views

Find order of elliptic curve

Given a prime $p$ such that $3$ does not divide $p-1$, what is the order of the elliptic curve over $\mathbb{F}_p$ given by $E(\mathbb{F}_p)=\{ (x,y) \in \mathbb{F}_p^2 | y^2=x^3+7 \}$ I thought if ...
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5answers
102 views

Can anyone prove the identity $\sum_{m=-\infty}^\infty (z+\pi m)^{-2} = (\sin z)^ {-2} $

I came across this identity in a paper on elliptic curves, and the proof wasn't provided. It really irked me, and I couldn't find an explanation anywhere else. Can anyone shed some light? ...
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2answers
58 views

About Mordell's Theorem (Elliptic Curves)

I've just finished the proof of Mordell's Theorem given in the book "Rational Points on Elliptic Curves " by Silverman. One of the key lemmas used in the proof of the theorem is: Let ...
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2answers
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Substitutions that transform Fermat Equations to Elliptic Curves

I was reading Chapter 1 of Elliptic Curves - Number Theory and Cryptography by Lawrence C Washington. He was considering Fermat equations $$a^4+b^4=c^4\text{ and }a^3+b^3=c^3.$$ For the 1st equation, ...
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1answer
55 views

How do i find all integers $y$ such that $y^3 = 3x^2+3x+7$, where $x$ is also an integer?

I have tried to find all integers $y$ such that $$y^3 = 3x^2+3x+7$$, where $x$ is also an integer but i didn't succed only i guess that no integer $y$ $x$satisfied that equation so i would like to ...