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

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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?
3
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1answer
79 views

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
49 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
55 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
80 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
48 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 ...
8
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1answer
149 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
96 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 ...
2
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1answer
45 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 ...
4
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2answers
65 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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0answers
68 views

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 ...
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1answer
47 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
votes
2answers
134 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
88 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
91 views

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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31 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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127 views

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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0answers
37 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 ...
7
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1answer
255 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 ...
2
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1answer
135 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 ...
3
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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 ...
4
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2answers
80 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 ...
2
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1answer
74 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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115 views

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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0answers
44 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 $F_v \in Gal(K^{ab}/K)$ ...
2
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1answer
43 views

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
30 views

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 ...
2
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1answer
82 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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19 views

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
39 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 ...
6
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1answer
143 views

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
34 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
109 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
75 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 ...
5
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2answers
118 views

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, ...
0
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1answer
62 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 ...
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0answers
89 views

The Frobenius Trace for an elliptic curve

Let E be an elliptic curve defined over $\mathbb{Q}$ (coeffs. there), and consider its $n-$torsion points in $\mathbb{C}$, $E(\mathbb{C})_{\text{tors}}[n]$. We know this group is isomorphic to ...
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1answer
89 views

Addition of points on elliptic curves over a finite field

I have found the following formulas for the coordinates of $P+Q$ given that $P = (x_{1}, y_{1})$ and $Q = (x_{2}, y_{2})$ are points on a general curve $y^2 = x^3 + ax + b$ over $\mathbb{R}$: $$P + Q ...
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2answers
200 views

Modular curves over finite fields

I'm looking for a detailed reference for modular curves over finite fields, such as $X(N)$, $X_1(N)$, and $X_0(N)$. There seems to be a lot of literature dealing with them over $\mathbb{C}$, but I'm ...
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175 views

Completion of the proof of theorem 3.3 in Dale Husemoller: Elliptic Curves

I want to read the proof of the following theorem: This is from p.35. But it is not complete there. There is written that: Can someone tell me where I can find the rest of the proof? Any other ...
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0answers
54 views

For which values of $k$ does: $ y^2 = x^3+(2^{2^k}+1)x$ have solutions in integers?

let $E_D$ be elliptic curve and $k$ is integer number $$E_D: y^2 = x^3+px. $$ When $p = 2^{2^k}+1$ is prime fermat . my question is :For which values of $k$ does:E.d $$ y^2 = x^3+px. $$ have ...
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1answer
106 views

Finding a stronger version of Cayley-Bacharach Theorem that applies in the case that the intersection multiplicities are not equal to $1$

Cayley–Bacharach theorem: Assume that two cubics $C_1$ and $C_2$ in the projective plane $\mathbb{P}^2$ meet in nine (different) points. Then every cubic that passes through any eight of the points ...
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1answer
70 views

Some questions on elliptic curves over finite fields

Let $E$ be an elliptic curve defined over $\mathbb{F}_q$. For a prime $\ell \neq q$, we have that the $\ell$-torsion subgroup $E[\ell] \cong (\mathbb{Z}/\ell \mathbb{Z})^2$. As can be easily seen, ...
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0answers
58 views

Can this relationship be expressed algebraically?

$\frac{\left(x-1\right)!+1}{x}=\frac{\left(y-1\right)!+1}{y}$ When I graphed it, I noticed that it bears a resemblance with the equation (which could of course be completely coincidental): ...
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2answers
73 views

Finding some rational points on elliptic curves

If I am considering an elliptic curve, for example $$y^2=x^3-2$$ $$\text{Edit: and } y^2=x^3+2$$ over $\mathbb Q$, how to find rational points? What possibilities do we have to calculate ...
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1answer
48 views

Elliptic curves, reduction map, $E_n$

Let $E$ be the elliptic curve and set $\phi: E(\mathbb{Q}_p) \rightarrow E(\mathbb{F}_p)$ to be the reduction morphism. Define $E_n := \{(x:y:z) \in \ker \phi | x/y \in p^n\mathbb{Z}_p\}$. I'm busy ...
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1answer
86 views

Questions on branch points on elliptic curve

So let $(E,p)$ be an elliptic curve over a field $k$ with a choice of $k$-valued point $p$. Then by Riemann-Roch, there are two global sections of $\mathcal{O}_{E}(2p)$ which gives a double cover of ...
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51 views

Elliptic curve over field $\mathbb{C}(\lambda)$

I have the following problems on my modular forms course final exam: Over field $\mathbb{C}(\lambda)$ equation $y^2 =x(x−1)(x−\lambda)$ defines an elliptic curve $E_{\lambda}$ with a base in ...
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2answers
67 views

Help with proving that the torsion subgroup of $y^2=x^3+x$ is $E(\mathbb{Q})_{tors} \cong \mathbb{Z}/2\mathbb{Z}$

Let $E: y^2= x^3 + x$ be an elliptic curve over $\mathbb{Q}$. I'm trying to prove that $E(\mathbb{Q})_{tors} \cong \mathbb{Z}/2\mathbb{Z}$. In order to do that, I've already shown that ...
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0answers
29 views

Complex elliptic surface with 24 $I_1$ fibers

Is a complex elliptic surface with 24 $I_1$ fibers always a K3 surface? Is ti possible to characterize a K3 surface in terms of the singular fibers of a given elliptic surface?