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

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3
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109 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 ...
2
votes
1answer
60 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 ...
0
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0answers
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 $F_v \in Gal(K^{ab}/K)$ ...
3
votes
1answer
39 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 ...
1
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1answer
29 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
votes
1answer
56 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 ...
4
votes
2answers
150 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 ...
4
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2answers
78 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
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 ...
0
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0answers
16 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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3answers
1k views

What is an elliptic curve, and how are they used in cryptography?

I hear a lot about Elliptic Curve Cryptography these days, but I'm still not quite sure what they are or how they relate to crypto...
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1answer
79 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 ...
0
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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 ...
2
votes
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? ...
3
votes
1answer
60 views

Pushing forward vector bundles on a plane curve via projection from a point

Let $C \subset \mathbb{P}^2$ be a smooth plane curve, $P \in \mathbb{P}^2$ is point not on $C$, consider projection from this point $$ \pi :\mathbb{P}^2 - \{P\} \to \mathbb{P}^1, $$ and restrict this ...
4
votes
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 ...
0
votes
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 ...
3
votes
0answers
46 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 ...
2
votes
1answer
91 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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0answers
53 views

Group Operation of points on a Montgomery elliptic Curve (project coordinates)

I was trying to implement the double and addition formulas for elliptic points on a Montgomery elliptic curve. I came across this weird thing which should definitely not be happening. I took a point ...
1
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1answer
56 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 ...
4
votes
2answers
139 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 ...
2
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1answer
53 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, ...
0
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0answers
50 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 ...
1
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1answer
51 views

hyperelliptic curve

Please help me to solve this question: Let $H$ be a hyperelliptic curve over $\mathbb{F}_{103}$ given by the equation $ y^2 = x^5+1$. let $J$ be the jacobian of $H$ defined over $\mathbb{F}_{103}$. ...
0
votes
0answers
53 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): ...
3
votes
1answer
42 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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2answers
54 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 ...
1
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1answer
58 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 ...
3
votes
1answer
61 views

Show that $Y^2-X^3\mid f$ if $f$ vanishes on the curve $C: (t^2,t^3)$, and determine what property of a field $k$ will ensure that the result holds.

Let $\phi: \mathbb{R^1}\rightarrow \mathbb{R^2}$ be the map given by $t \mapsto (t^2,t^3)$; prove directly that any polynomial $f\in \mathbb{R}[X,Y]$ vanishing on the image $C=\phi(\mathbb{R^1})$ is ...
4
votes
2answers
59 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
42 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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1answer
1k views

Does an elementary solution exist to $x^2+1=y^3$?

Prove that there are no positive integer solutions to $$x^2+1=y^3$$ This problem is easy if you apply Catalans conjecture and still doable talking about Gaussian integers and UFD's. However, can this ...
3
votes
1answer
49 views

Weil pairing, cyclotomic field in division field and determinant map for ell curve and abelian variety

Question 1: If $E/K$ is an elliptic curve defined over a number field $K$, then the Weil Pairing gives me that $K(\mu_n) \subseteq K(E[n](\bar{K}))$. If i identify $Gal(K(\mu_n)/K)$ with a subgroup ...
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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?
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2answers
102 views

Points on elliptic curve over finite field

Find the points on the elliptic curve $y^2 = x^3 + 2x + 2$ in $\mathbb F_{17}$. Do I have to guess a first point and then use an algorithm to spit out all other points?
7
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1answer
191 views

Integral relations in Fricke and Klein

Can someone please explain how Fricke and Klein obtain the integral relationa stated at the top of p. 34 in this book? The entire book can be previewed on Google Books. It is an old book and I do not ...
3
votes
1answer
54 views

Real Lie groups and elliptic curves

Let $f:A\to A'$ be a morphism of elliptic curves over the real numbers $\mathbb R$. It canonically induces a morphism $f(\mathbb R): A(\mathbb R)\to A'(\mathbb R)$ between the sets of real points, ...
2
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1answer
43 views

Solving equations in $\mathbb{Z}_3$ with Hensel's Lemma

Further to the post here, I'm trying to find the $n \in \mathbb{Z}$ such that there is a solution to the equation $$ x^3 +3x+y^3+3y=n$$ in $\mathbb{Z}_3$. Now, I've been able to show that in the ...
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1answer
60 views

Why so few complex multiplication

Let's take I lattice $\Lambda$ and $\alpha\in\Lambda$. Then we have $\alpha\Lambda\subseteq\Lambda$ so $z\mapsto \alpha z$ induces an isogenie $\mathbb{C}/\Lambda\to\mathbb{C}/\Lambda$ wich has no ...
4
votes
2answers
116 views

Can an elliptic curve have discrimant one?

Can the discriminant, $4a^3 +27b^2$ of an elliptic curve $$E: y^2=x^3+ax+b$$ be equal to 1. I believe that this should not be possible otherwise the curve would have good reduction at all primes $p$, ...
3
votes
2answers
24 views

Faster scalar multiplication over an elliptic curve by hand?

For an elliptic curve $y^2=x^3+ax+b$, I have $a=1, b=1, G=(3,10)$ private key of User $B$ as $4$. To calculate his public key, I have the formula: $Pb=nb \times G = 4(3,10)$. This makes my ...
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1answer
38 views

Dividing elliptic curve point on integer

Can we solve the equation $nP = Q$, where $P$, $Q$ is a rational poins on elliptic curve ($P$ is unknown), and $n$ is integer?
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1answer
43 views

Group on elliptic curve points

Let's we have an elliptic curve (EC). Is it possible to construct group $G$ acting on the points of an EC with this property: if $P$ is a rational point on EC then $G(P)$ is also is a rational point? ...
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0answers
50 views

How I can determine the number of integral solutions of the equation of the elliptic curve?

Is there a general law to determine how many integral solutions of the Equation of the form (elliptic curve): $y²=x^{3}+ax+b,\ \ \ a,b \in \Bbb R$. Thank you for any help .
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0answers
25 views

Computing the order of a divisor in the Jacobian of a hyperelliptic curve.

Given a hyperelliptic curve of genus $g$, of equation $H: y^{2}+h(x)y=f(x)$ and defined over the finite field $\mathbb{K}$, how does one compute the order of a (reduced) divisor defined over ...
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vote
1answer
29 views

Number of points on an elliptic curve and it's twist over $\mathbb{F}_p$.

I have another probably very trivial question about elliptic curves. This wikipedia article gives the following formula $|E|+|E^d|=2p+2$ where $E$ is an elliptic curve over $\mathbb{F}_p$ and $E^d$ is ...
2
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1answer
44 views

Pole Order of Weierstrass Coordinates

I'm trying to understand a proof in Silverman's The Arithmetic of Elliptic Curves. Background: For an elliptic curve $E$, $x, y \in K(E)$ such that $\phi = [x, y, 1]: E \rightarrow C$ is a basepoint ...
5
votes
0answers
149 views

What is stopping every Mordell equation from having a [truly] elementary proof?

The Mordell equation is the Diophantine equation $$Y^2 = X^3-k \tag{1}$$ where $k$ is a given integer. There is no known single method — elementary or otherwise — to solve equation $(1)$ for all $k$, ...
3
votes
2answers
84 views

Elliptic curve $y^2= x^3 + x$ over the finite field $\mathbb{F}_p$ with $p \geq 3$.

Consider the elliptic curve $$E: y^2= x^3 + x$$ over the finite field $\mathbb{F}_p$ with $p \geq 3$. I want to show that $|E(\mathbb{F}_p)| \equiv 0 \mod 4$. I know that, if $p \equiv 3\mod 4$, ...