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

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3
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1answer
159 views

Elliptic curve- Component of point

If $E/ \mathbb{Q}$ elliptic curve in the general Form of Weierstrass and $P=(x,y)$ a rational point of it, show that the first coordinate of the point $2P$ is $$ ...
4
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0answers
101 views

A cubic equation: $u^3−2u^2−2v^3−20v^2+16v=0$

Update (Dec. 22): I have already solved this question with Magma. Recently, I read a paper [1] and saw the following equation: $$u^3−2u^2−2v^3−20v^2+16v=0.$$ The author then got a Weierstrass ...
0
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0answers
102 views

Book/lecture notes on algebraic curves

Although there surely is plenty of references on MSE about algebraic curves, my need are very specific and so I will open this topic anyway. I follow this year a course on (hyper)elliptic curves ...
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2answers
67 views

Elliptic curves find points with rational coordinates

The elliptic curve $y^2=x^3+3x+4$ has points O,(-1,0) and (0,2). Find five more points with rational coordinates. The answer to this example gives: (0,-2) (5,-12) (5,12) (71/25,744/125) and ...
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4answers
189 views

Elliptic curves (sum and multiply)

I was wondering if someone could give me some resources on elliptic curve cryptography. Specifically I need to know how to do something like: $y^2=x^3-x+1$ compute $(0,1)⊕(1,1)$ or $y^2=x^3+x^2-x$ ...
-2
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1answer
67 views

Elliptic curves 2P, 3P

How do I compute 2P, 3P etc? ex: $y^2=x^3+4xmod7$ and I have to compute the order of (2,3)=P and my example says 2P =(0,0) 3P=(2,4) but I don't know how to get these answers?
2
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2answers
76 views

Elliptic curves as cubics as discussed in Ravi Vakil's notes

I was reading the section of Ravi Vakil's Algebraic Geometry notes where he discusses elliptic curves. If we let an elliptic curve be $(E,p)$ (Where $p$ is the distinguished point), we have ...
3
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1answer
108 views

Multiplicity of intersection between tangent and elliptic curve

Doubling a point (adding it to itself) on an elliptic curve is done by taking the tangent to the point and calculating the other point where the line intersects the curve. That point is then reflected ...
1
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1answer
31 views

Elliptic curves: Can I replace a coordinate with any modularly equivalent number?

I have a point (x, y) in an elliptic curve group. Suppose y is negative. Can I rewrite it as a positive number if that positive number is equivalent to y (modulo the characteristic of the group)? ...
8
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3answers
372 views

What does the Tate module of an elliptic curve tell us?

I started studying elliptic curves, and I see that it is rather common to take the Tate module of an elliptic curve (or, of the Jacobian of a higher genus curve). I'm having a hard time isolating the ...
3
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1answer
126 views

Definition of a Elliptic curve

I've seen two different definitions of an elliptic curve. 1) The first one being that it is a nonsingular projective curve of genus 1. 2) The other definition nonsingular projective curve of ...
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1answer
82 views

Let $y^2 = x^3 + Ax + B$ be a curve and $y = m(x - x_1) + y_1$ tangent at $x_1$. Why is $x_1$ then a double root?

Suppose we have a function $y^2 = x^3 + Ax + B$ which we differentiate implicit to find $$\frac {dy} {dx} = \frac {3x^2 + A} {2y}$$ Now suppose we know a point $(x_1,y_1)$ on the curve. Define $$y = ...
7
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3answers
761 views

References for elliptic curves

I just finished reading Silverman and Tate's Rational Points on Elliptic Curves and thought it was very interesting. Could any of you point me to some more references (ex. books, articles) on ...
2
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2answers
53 views

Does the Hasse inequality fail for supersingular elliptic curves?

For supersingular elliptic curve $E: y^2+y=x^3 + 2x$ over $\mathbb{F}_{27}$, $\#E\left(\mathbb{F}_{27}\right) = 55$ but $|\#E(\mathbb{F}_{q}) - (q+1)| \leq 2\sqrt{q} \iff 18 \leq \#E(\mathbb{F}_{27}) ...
1
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1answer
61 views

Solutions of elliptic curve in finite field

If I take the following elliptic formula over a finite field of size $17$: $$y^2 = x^3 + 2x + 3$$ The solutions for $x = 2$ would be $7$ and $10$. Because $7^2=49$ and $49 \equiv 15 \bmod 17$ ...
3
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0answers
63 views

Describing generators for the fundamental group of an elliptic curve given by an equation

Say you're given an equation in the form $y^2 + a_1xy + a_3y = x^3 + a_2x^2 + a_4x + a_6$. If the $a_i$'s are complex numbers, the subset $E^*\subset\mathbb{C}^2$ satisfying this equation is a ...
0
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1answer
38 views

Group law, cubics and Lie group

Let $C$ be a smooth complex cubic in $CP^{2}$. We know that there is a group structure by using the intersection of projective lines (cf. Ried, Undergraduate AG, Section 2), which is really different ...
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0answers
174 views

3D equation of a cone-like shape

Imagine there are two parallel planes (base plane and plane1) in the following image: There is one point on the base plane and there are several points on the plane1. The positions of these points ...
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0answers
44 views

Does this family of complex elliptic curves have a nontrivial section?

Start with the product $\mathbb{C}\times\mathcal{H}$ ($\mathcal{H} = $ upper half plane). Define an action of $\mathbb{Z}^2$ on the left by $(m,n)\cdot(z,\tau) := (z + m\tau + n,\tau)$. Then the ...
0
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1answer
71 views

graduate level introduction to elliptic curve cryptography

I am looking for a good modern book / lecture-notes about elliptic curve cryptography. Does anyone have good recommendations?
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36 views

Computing cohomology of the sheaf $\mathcal{End}(T_{\mathbb{P}^2})$ restricted to a curve

Characteristic of the basic field is zero in this question. Let $E \subset \mathbb{P}^2$ be a smooth elliptic curve. Let $\mathcal{F}$ be the vector bundle on $E$ obtained as restriction to $E$ of the ...
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0answers
17 views

Constrained determination of an elliptic curve with marked points

I am trying to determine some equilibrium position of electrostatic charges on Riemann surfaces, and I was wondering if the following problem was a classical type of problem; Let $z_1,...,z_n$ be n ...
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0answers
96 views

Hasse's theorem on elliptic curves over finite fields

Suppose $\mathcal{E}$ is an elliptic curve defined over $\mathbb{Q}$, Then Hasse's theorem states that for any large characteristic $p$ there exists an algebraic number $\lambda_p$ of modulus ...
0
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1answer
23 views

Specific cartesian coordinates of an ellipse

I want to do the following: 1.) Ask user for the vertical and horizontal distances of the ellipse 2.) With this information calculate the circumference 3.) Divide the circumference by the closest ...
4
votes
2answers
119 views

Concrete and elementary applications of modular forms to elliptic curves

What are some useful facts/algorithms for elliptic curves that can be obtained (proved completely) using the theory of modular forms without heavy machinery? It's often been asked what elementary ...
2
votes
1answer
62 views

Clarification of the proof of **Proposition 1.5** in Silverman

I'm studying the proof of Proposition 1.5 on silverman "The Arithmetic of Elliptic Curves" : Proposition: Let $E$ be an elliptic curve. Then the invariant differential $\omega$ associated to a ...
0
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1answer
34 views

Multiply point by scalar in elliptic curve group

I'm trying to understand how to multiply a point by a scalar to get a point in elliptic curve cryptography. Here's an example from my textbook. The group is E257(0, -4). That's shorthand for y2 = x3 ...
3
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0answers
59 views

Klein's invariant and negative discriminant

Let $J = J(\tau)$ be Klein's invariant and let $0 < k < 1$ be the elliptic modulus. It is known that $$J = \frac{4}{27} \frac{(1 - \lambda + \lambda^2)^3}{\lambda^2 (1 - \lambda)^2},$$ where ...
2
votes
1answer
71 views

Punctured Elliptic Curve

I've come across the word "punctured elliptic curve" here and there, but none of the basic texts on the topic (Husemoller, Silverman) define or mention it. What point is removed from the curve (the ...
2
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1answer
43 views

How long does it take to double and add points on an elliptic curve?

I have some problems understanding how many multiplications it takes to add or double points on an elliptic curve in Weierstrass form. This link tells that it's 11 and 14, but I don't quite understand ...
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93 views

Elliptic Curves

I want some clarification regarding some concept in elliptic curves. In many papers I have seen that, let $E:y^2=x^3+Ax+B $ be an elliptic curve if $L(E,1) $ (corresponding L-function at s=1) is ...
3
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1answer
70 views

Elliptic curves with twists of zero rank

I am new to field of Elliptic curves. When I was seeing some papers related to this area I have come across elliptic curves having quadratic(some times cubic ) twists with zero rank. What is the ...
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1answer
33 views

How I can calculate the algebraic rank of $C(ℚ)$

Let us consider the elliptic curve $C$ over $ℚ$ in Weierstrass form $$C:y²=x³+1$$ How I can calculate the algebraic rank of $C(ℚ)$.
3
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1answer
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Generators of elliptic curves?

How to find generators to group $E(\mathbb Q) $ of following elliptic curves $E:y^2=x^3-198 $, $E:y^2=x^3-122 $. Thank you in advance.
2
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1answer
70 views

Chapter II Example 3.3 -p.28 Silverman

I have a question about Chapter II Example 3.3 -p.28 in Silverman "Arithmetic of Elliptic Curves". I feel like I'm misreading it and would like clarification. Let $K$ be a field such that ...
11
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2answers
346 views

Why can't elliptic curves be parameterized with rational functions?

Background: For our abstract algebra class, we were asked to prove that $\mathbb{Q}(t, \sqrt{t^3 - t})$ is not purely transcendental. It clearly has transcendence degree $1$, so if it is purely ...
4
votes
1answer
128 views

Rational map of a curve to an elliptic curve

If I have a curve given by $$ y^2 = (x^3-1)(x^3-a), $$ how do I find out if there is a rational variable transformation $y=y(s,t)$, $x=x(s,t)$ that maps this curve onto an elliptic curve of the form ...
1
vote
1answer
31 views

Approximating the Rank

Let $$L(C,s)=\sum_{n=1}^\infty \frac{a_n}{n^s}$$ be the Dirichlet series of the Hasse--Weil L-function of an elliptic curve $C$ over $ℚ$. The modularity theorem implies that $L(C,s)$ is the ...
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0answers
32 views

Ample non-flex on elliptic curve

Say $E$ is a plane cubic, and $p$ is a point on $E$. Riemann-Roch tells us that $\mathcal O_E(3p)$ is very ample. If $p$ is a flex, it's easy to write down the three sections giving an embedding of ...
3
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1answer
57 views

The definition of an elliptic curve?

I've seen two different definitions of an elliptic curve. The first one being that it is a cubic curve of the form $y^2=x^3+ax^2+bx+c$, where all the (complex) roots are different. The other ...
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0answers
86 views

Proof of elliptic curves being an abelian group

What are some simple proofs that the points on an elliptic curve form an abelian group under addition? I am mostly looking for proofs of closure and associativity, since the other three requirements ...
4
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1answer
164 views

solving $x^3-2y^3=1$ using cubic number field

I am trying to solve the diophantine equation $x^3-2y^3=1$ using $\mathbb{Q}(\sqrt[3]{2}).$ I've read this link: Solve $x^3 +1 = 2y^3$ The following is what i have tried: Finding all integer ...
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1answer
62 views

Is the union of all elliptic curves $\mathbb{R}^2$?

An elliptic curve could be written as $$y^2 = x^3 + a x + b \;.$$ Q1. Is it the case that every point $p \in \mathbb{R}^2$ lies on some elliptic curve? Q2. And what is the natural ...
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1answer
93 views

Quantities $g_2$, $g_3$, $\Delta$

This question is somewhat related to this one. Let $\lambda$ be the modular lambda function. Greenhill (Elliptic Functions, p. 57) states that we may put $$g_2 = \frac{1 - \lambda + \lambda^2}{12}, ...
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0answers
81 views

Finding a point on an elliptic curve

I have an elliptic curve with the equation $ y^2 = x^3 + ax + b $ in modulo p, where p is prime. I also have a point G on that curve. How can I find another point that isn't a multiple of G? I ...
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0answers
32 views

The correct formula for lambda when point doubling?

When doubling a point on an elliptic curve, we use $\lambda$. But the equation I found in my book(Silverman and Tate, Rational Points on Elliptic Curves) isn't the same as the one I found when looking ...
4
votes
1answer
72 views

First problem in Silverman's Arithmetic of Elliptic Curves

I started working through Silverman's Arithmetic of Elliptic Curves. For some reason it looks like the first problem in the first chapter is the hardest problem in the whole chapter or I'm completely ...
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vote
2answers
112 views

Constructing a meromorphic function

I need help with the following problem. "Let $C : y^2 = x^3 − 5x^2 + 6x$ be a cubic curve with the standard group law. Find a meromorphic function on $C$ having the pole of order two at ...
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1answer
72 views

Why is reduction modulo $p$ a group homomorphism on Elliptic Curves?

I am reading A. Knapp's book on elliptic curves right now. In Proposition 5.6 the author wants to prove that the reduction map (modulo $p$, where $p$ does not divide the discriminant) of an elliptic ...
3
votes
2answers
142 views

Elliptic curves on a K3 surface

Let $X$ be an elliptic K3 surface. Let $\alpha$ be a smooth curve of genus $\geq3$. Define $$d(\alpha)=\min\lbrace \epsilon\cdot \alpha \ | \ \epsilon \mbox{ is an elliptic curve on } X \rbrace, $$ ...