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

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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 $\mathrm{...
6
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1answer
153 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
32 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 $L$-...
3
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1answer
69 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 ...
3
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0answers
34 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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0answers
73 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 $\...
27
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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 ...
4
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1answer
73 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
192 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 ...
3
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1answer
64 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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0answers
120 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 ...
4
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1answer
97 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}, \...
2
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1answer
87 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 ...
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0answers
44 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 ...
5
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1answer
203 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 ...
1
vote
1answer
113 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 ...
5
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1answer
218 views

Cube of an integer

$\frac{x}{y}+\frac{y}{z}+\frac{z}{x}=k$ and $x, y, z, k$ are integers. Prove that $xyz$ is cube of some integer number. I was wondering about giving a parametrization for the rational points on ...
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0answers
54 views

Quick explanation of $\Gamma \tau$ notation?

I hope you can help me by quickly explaining the following notation: $\Gamma \tau$. This notation is encountered in A First Course in Modular Forms by Fred Diamond and Jerry Shurman (love the book by ...
1
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1answer
61 views

class number of pure cubic fields and elliptic curves

I want to find generators to Mordell Weil group of the Elliptic Curve $y^2=x^3−6321363052$ and class number of $\mathbb Q(\sqrt[3]{6321363052})$. Some suggestions such as algorithm or softwares will ...
3
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0answers
66 views

Elliptic curves as $\mathbb{C}^*/\mathbb{Z}$

I apologize in advance if my question is rather trivial, but i have trouble understanding a basic fact about elliptic curves.. I have always wrote an elliptic curve $E$ as $\mathbb{C}/\Lambda$, where $...
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0answers
82 views

Computation of the 2-torsion group of an elliptic curve

I have some troubles solving the following problem: Let $E$ be the elliptic curve $E:y^2+2y=x^3+x+9$ over $\mathbb{F}_{16}$. Compute the 2-torsion group $E[2]$, i.e. find all the points of order $2$ ...
0
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1answer
135 views

Compute principal divisor for a rational function on a curve

During the lecture we defined the principal divisor of a rational function on a smooth curve as it follows: Consider the smooth curve $C\subseteq\mathbb{P}^2$. Take $g\in{K(C)^*}$. Then the principal ...
9
votes
2answers
122 views

order of an elliptic curve

I have found that the curve given by $x^3+x+1=y^2$ over $\mathbb{F_5}$ has 9 points. Now I am supposed to find the number of points of the same curve on $\mathbb{F}_{125}$. Using Hasse and the fact ...
2
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1answer
214 views

Looking for help with this elementary method of finding integer solutions on an elliptic curve.

In the post Finding all solutions to $y^3 = x^2 + x + 1$ with $x,y$ integers larger than $1$, the single positive integer solution $(x,y)=(18,7)$ is found using algebraic integers. In one of the ...
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2answers
108 views

Quadratic twist of Elliptic curves with complex multiplication

Suppose $E/\mathbb{Q}$ is an elliptic curve that has complex multiplication by $\mathcal{O}_K$, where $K=\mathbb{Q}(\sqrt{D})$, for $D<0$ and squarefree. In "The main conjectures of Iwasawa ...
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1answer
70 views

Seeking graphics of elliptic curves as surfaces

Is there any place to go on line for good graphics of how a complex elliptic curve sits as an affine curve in $\mathbb{C}^2$? The mathematics is well discussed in Drawing elliptic curve and Is the ...
2
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1answer
60 views

Surjectivity of p-adic representation

Let $E/\mathbb{Q}$ be an elliptic curve. For a prime $p$, we have the mod $p$ representation \begin{equation*} \bar{\rho}_{E,p}: G_{\bar{\mathbb{Q}}/\mathbb{Q}} \rightarrow Aut(E[p]) \end{equation*} ...
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0answers
120 views

Non linear least square ellipse fitting

I am trying to find a Non linear leasts squares ellipse fit for a set of 100 data points data points $(x,y)$. Now i have found the values of $A,B,C,D,E,F$ according to the conical equation of the ...
0
votes
2answers
107 views

defining the group law on elliptic curves in general

Let $k$ be an arbitrary field and $C \subset \mathbb{P}^2(k)$ an elliptic curve. In order to define the group law on $C$ we need to establish some geometric facts first, e.g. Any line intersects $C$ ...
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1answer
470 views

The Moduli Stack of Elliptic curves - What is it?

I have often heard the words "Moduli Stack of Elliptic Curves", but I have nowhere found a from-scratch definition of this object. I do understand the motivation: There are cusps in the moduli space ...
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1answer
44 views

Image of the p-Frobenius endomorphism under a mod p Galois representation.

Let $E/\mathbb{Q}$ be an elliptic curve and $p$ a prime such that $E$ has ordinary redcution at p. Further, let $$\rho_{E,p}:{\rm Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\to {\rm GL}_2(\mathbb{Z}/p\...
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2answers
69 views

Compendium of elliptic curves?

does anyone know where I can find a collection of elliptic curves and their integral solutions? EDIT: Removed additional useless info. Thanks!
3
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1answer
135 views

Rank of Elliptic Curves

Recently, I have heard of some heuristics that would suggest that the rank of elliptic curves are bounded (specifically in the congruent number family). I always though that the best way to prove ...
2
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2answers
188 views

How to prove $~\sqrt{3}\sqrt{4a^{3}-1}~$ isn't an integer?

I'm trying solve: $~a^3 + b^3 = c^3~$ has no nonzero integer solutions. If $~(c−b)=1~$ then $~c^3-b^3=3c^2-3c+1=a^3,~$ from Wolframalpha get: $$ c = \dfrac{3- \sqrt{3}\sqrt{4a^{3}-1}}{6} \\ c = \...
2
votes
1answer
98 views

Computing the trace of the image of Frobenius

Let $E/\mathbb{Q}$ be an elliptic curve, $p$ and $q$ distinct primes and $e$ a positive integer. Fixing a basis for the $p^e$ torsion, we get a natural Galois representation $$\rho_{E,p^e}:{\rm Gal}...
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1answer
40 views

Surjective $p$-adic representation implies trivial $p$-primary part.

Let $E/\mathbb{Q}$ be an elliptic curve. We know that by Serre in the non-CM case, for $p\geq5$, $$\rho_p:Gal(\bar{\mathbb{Q}}/\mathbb{Q})\rightarrow Aut(T_p(E))$$ is surjective iff $$ \bar{\rho}_p:...
3
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1answer
51 views

Can you have a nontrivial automorphism of an elliptic curve $E/S$ which when restricted to a geometric fiber is the identity?

Ie, let $E/S$ be an elliptic curve over some scheme $S$. Is it possible to have an automorphism $\alpha$ of $E$ over $S$ such that for some geometric point $s\in S$ its pullback to $E_s$ is the ...
0
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1answer
21 views

Tangent line at $x_1$ to polynomial curve $p(x)$ of degree at least $2$ implies $x_1$ is a double root of $p(x) - p^{'}(x_1)(x-x_1)$?

Tangent line at $x_1$ to polynomial curve $p(x)$ of degree at least $2$ implies $x_1$ is a double root of $p(x) - p^{'}(x_1)(x-x_1)$ ?. Suppose I have a polynomial function $p(x): \mathbb R \...
0
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1answer
174 views

Third point of elliptic curve $E: y^2 = x^3 + Ax + B$ given points $P_1=(x_1,y_1), P_2=(x_2, y_2)$ on $E$ (Weierstrass equation).

Third point of elliptic curve $E: y^2 = x^3 + Ax + B$ given points $P_1=(x_1,y_1), P_2=(x_2, y_2)$ on $E$ (Weierstrass equation). Assume $x_1 \neq x_2$. I create the straight line $y = m(x-x_1) + ...
3
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2answers
310 views

explicit example of computing ray class field for imaginary quadratic?

Given an imaginary quadratic number field K, we can get its ray class field mod some ideal $\mathcal{m}$ by adjoining the j-invariant of an elliptic curve with complex multiplication given by $\...
2
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1answer
353 views

Flex point on an elliptic curve

I have just started working through Pete Clark's elliptic curve notes, which are available here: http://math.uga.edu/~pete/EllipticCurves.pdf Early on, in section 2.1 on page 6, it is shown that the ...
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1answer
1k views

How to find primitive point on an elliptic curve?

Reading about Elliptic curve cryptography, i came across primitive point's or generator point's but found nothing on how to generate such points any help would be appriciated.
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1answer
242 views

Relating ramification index of a map of curves to degree of vanishing

I am little confused about explicitly computing ramification index and relating it to degree of vanishing a polynomial. In particular I have the following example (when trying to prove the genus ...
2
votes
1answer
167 views

Solutions to a diophantine equation

I tried to find integer solutions to the following diophantine equation $$x^3 - 3y^3 + 5z^3 - 3xy^2 + 3x^2y + 9xz^2 + 7x^2z + 3yz^2 - 3y^2z + xyz = 0$$ but was unable to do so. I suspect that there ...
2
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0answers
94 views

Torsion points on elliptic curves, $E^1$

Let $E$ be an elliptic curve over $\mathbb{Q}_p$ and let $E(\mathbb{Q}_p) \supset E^0(\mathbb{Q}_p) \supset E^1(\mathbb{Q}_p) \supset \cdots$ be its $p$-adic filtration, where $E^n(\mathbb{Q}_p) = \{P ...
0
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1answer
36 views

How come $f(0) = 0$ in $\mathbb C/L$?

How come $f(0) = 0$ in $\mathbb C/L$? Does anyone know it? Your help will be appreciated. This is taken from the text "Rational Points on Elliptic Curves" by Tate and Silverman.
6
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1answer
214 views

Lenstra's Elliptic Curve Algorithm

I am currently trying to understand Lenstra's Elliptic Curve Algorithm for factoring integers. As a source I use "Rational Points on Elliptic Curves" by Joseph H. Silverman and John Tate. They ...
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1answer
133 views

Multiple points on an elliptic curve

I have given the following elliptic curve $E:F(x,y) = 0$: (Where $F(X,Y) := Y^2 + a_1XY + a_3Y - X^3 - a_2X^2 - a_4X - a_6$ with $a_1 = -1.5, a_2 = 3, a_3 = 1, a_4 = 0.5, a_6 = -1.5$.) The curve $...
2
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1answer
77 views

$p$-adic numbers and projective coordinates

Let $E/\mathbb{Q}_p$ be an elliptic curve and let $E^0(\mathbb{Q}_p)$ denote its nonsingular points. We accept that $E^0(\mathbb{Q}_p)$ is a subgroup of $E(\mathbb{Q}_p)$. Then let $\overline{E}^\text{...
2
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1answer
78 views

elliptic curves over $\mathbb{Q}_p$

Let $E: Y^2 = X^3 + AX + B$ be an elliptic curve over $\mathbb{Q}_p$, i.e. $A,B \in \mathbb{Q}_p$ and $4A^3 + 27B^2 \neq 0$. Then, according to page 47 of Cassels' Lectures on Elliptic Curves, if $(x,...