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

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5
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77 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. ...
2
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0answers
29 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 ...
4
votes
0answers
109 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
votes
1answer
61 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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0answers
45 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
votes
2answers
45 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
votes
1answer
51 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 ...
3
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0answers
108 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 ...
0
votes
0answers
40 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
37 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
vote
1answer
28 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
48 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 ...
0
votes
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 ...
0
votes
1answer
35 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
votes
1answer
74 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
votes
1answer
22 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
99 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? ...
4
votes
2answers
54 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 ...
4
votes
2answers
76 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
votes
1answer
51 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
36 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 ...
1
vote
1answer
52 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
132 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 ...
3
votes
2answers
135 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 ...
0
votes
0answers
44 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 ...
2
votes
1answer
88 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 ...
2
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1answer
50 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
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): ...
2
votes
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 ...
3
votes
1answer
41 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 ...
1
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1answer
57 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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0answers
39 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 ...
4
votes
2answers
57 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 ...
1
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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?
2
votes
1answer
40 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 ...
3
votes
1answer
52 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, ...
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1answer
49 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 $\mathbb{F}_{103}$. ...
2
votes
1answer
29 views

Elliptic Curve $E/\mathbb{Q}$ with $\Delta_E^{1/3}$ a root of defining cubic

Consider the elliptic curve $$E'\colon y^2 = g(x) = x^3 + \frac{1}{432} .$$ One can check that the discriminant of $E'$ is $-1/432$, that $E'$ has complex multiplication, and that $(-1/432)^{1/3} = ...
1
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1answer
59 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 ...
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0answers
44 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 ...
4
votes
2answers
115 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
1answer
46 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 ...
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
42 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? ...
0
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0answers
49 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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vote
0answers
22 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 ...
1
vote
1answer
27 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
votes
1answer
41 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 ...
3
votes
2answers
80 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 $p \equiv 3\mod 4$, ...