For questions regarding elliptic curves. Questions on ellipses should be tagged [conic-sections] instead.
0
votes
1answer
70 views
Bound of point's order on elliptic curve
For a given elliptic curve over a finite field and a point $P$ on that curve, how can we bound its order (integer $k$, such that $k*P=O$).
1
vote
1answer
87 views
Determining elliptic curve's parameters from addition procedure
Given a procedure that adds two points on an unknown elliptic curve, is it possible to determine curve's parameters, treating this procedure as a black box?
We are given two points on this curve $P$ ...
1
vote
1answer
253 views
Elliptic curve point addition
Are there any elliptic curves, that require computing GCD for point addition? I've an algorithm, that apparently adds two points on an elliptic curve, but it uses GCD, which is strange, because I ...
3
votes
1answer
494 views
Intuition and Stumbling blocks in proving the finiteness of WC group
After reading many articles about the Tate-Shafarevich Group ,i understood that "in naive perspective the group is nothing but the measure of the failure of Hasse principle,
and coming to its ...
6
votes
2answers
241 views
The Néron-Tate canonical height on elliptic curves
I have been trying to understand the Néron-Tate global canonical height of algebraic points on elliptic curves.
Let $K$ be a number field, $E$ an elliptic curve (over $\mathbb{Q}$, say), and $E(K)$ ...
3
votes
1answer
124 views
Finding a pencil of elliptic curves parametrized by a given modular surface
The following is an attempt to formulate a couple of questions which have been lurking in the back of my mind for a while. I'm sorry if this is long, or if my terminology is not correct, or if my ...
1
vote
0answers
177 views
About How to Elliptic Curve Equation and Discriminant
I am study public key cryptology and interested in elliptic curve cryplogical algorithms. I have some problem about elliptic curve equation. First I can't find inter process and transformation steps ...
1
vote
1answer
144 views
How to nicely extend finite field?
I'm working on an implementation of Miller's algorithm that computes the Weil pairing (elliptic curves, cryptography). In order to do that, I have to implement finite fields.
So far I have managed to ...
1
vote
0answers
63 views
For an elliptic curve E, does there exist a cofinite Fuchsian group without elliptic elements with quotient E minus a finite subset
Let $E$ be a compact Riemann surface of genus 1, i.e., an elliptic curve.
Let $P$ be the identity element of $E$.
Question 1. Does there exist a cofinite Fuchsian group (or a Fuchsian group of the ...
2
votes
1answer
126 views
Poincaré Residue Theorem
Can anyone point me to a reference which talks about periods of elliptic curves and the Poincaré Residue Theorem, hopefully one which uses this residue theorem to explicitly write out the period?
3
votes
1answer
132 views
Weierstrass Equation and K3 Surfaces
Let $a_{i}(t) \in \mathbb{Z}[t]$. We shall denote these by $a_{i}$. The equation $y^{2} + a_{1}xy + a_{3}y = x^{3} + a_{2}x^{2} + a_{4}x + a_{6}$ is the affine equation for the Weierstrass form of a ...
8
votes
1answer
119 views
Prove that a holonomic (p-recursive) difference equation returns only integral values
Consider the recurrence given by
$(n+1)^2 a_{n+1} = (9n^2+9n+3)a_n-27n^2 a_{n-1}$
$a_0 = 1, a_1 = 3$.
Clearly, $a_n$ is rational, but unexpectedly, the recurrence seems to output only integral ...
4
votes
1answer
295 views
Number of 3-torsion points on an elliptic curve
If we take our elliptic curve over $K$ to be the zero set of
$$
F(X_1, X_2, X_3) = X_2^2 X_3 - (X_1^3 + AX_1X_3^2 + BX_3^2),
$$
which is in projective form with $X = X_1, Y = X_2, Z=X_3$, then I ...
2
votes
1answer
365 views
Weierstrass Form of Elliptic Curve
One can put every cubic curve into Weierstrass form, how unique is this form?
1
vote
1answer
147 views
Intuition behind elliptic curves and $K$-rational points
I find myself becoming confused whenever I try to think about this. In the following, $K$ is a field.
An elliptic curve $\mathcal{C}$ is defined to be a nonsingular projective cubic curve over $K$, ...
2
votes
1answer
251 views
Birationally equivalent elliptic curves
I encountered a question about showing that the curve
$$
y^2 = x^4 + a_3 x^3 + a_2x^2 + a_1x + a_0, \qquad\qquad(1)
$$
where $a_i \in \mathbb{Q}$, can be birationally transformed over $\mathbb{Q}$ ...
0
votes
1answer
92 views
Maximal small lattice points of an elliptic curve
The elliptic curve $-4 x^3 + 4 x^2 y + 16 x - y^3 + 9 y$ goes through $21$ integer points in the range $-9$ to $9$. Is that the maximum?
6
votes
2answers
216 views
Is the real locus of an elliptic curve the intersection of a torus with a plane?
In Lawrence Washington's book Elliptic Curves: Number Theory and Criptography I read that if $E$ is an elliptic curve defined over the real numbers $\mathbb{R}$ then the set of real points ...
1
vote
1answer
191 views
Modular functions and elliptic functions
Does anybody know of an equation formally equating modular functions and elliptic functions similar to Euler's equation for exponential and trigonometric functions?
Any advice much appreciated.
...
5
votes
2answers
160 views
Injection of $E(\mathbb{Q})_{\text{tors}}$ into $\tilde{E}(\mathbb{F}_p)$?
I'm looking at Example VII.3.3.3 (p.193, 2nd ed.) of Silverman's The Arithmetic of Elliptic Curves. We have the elliptic curve $E:y^2=x^3+x$, with discriminant $\Delta=-64$, so there is good reduction ...
4
votes
1answer
91 views
Cokernel of morphism of Tate module of elliptic curves
Let $K$ be a field, and $\phi: E_1\to E_2$ be an isogeny of elliptic curves over $K$. Given a prime $\ell$ different from the characteristic of $K$, $\phi$ induces an injection $T_\ell \phi: T_\ell ...
4
votes
2answers
96 views
Bound for number of points on surface over $\mathbb{F}_p$
I know of the bound for the number of points on an elliptic curve over a finite field:
$$|\# E(\mathbb{F}_q) - q - 1| < 2\sqrt{q}$$
where this includes the point at infinity. I have been told that ...
4
votes
1answer
136 views
Extend an holomorphic function defined on a torus
Suppose we have an holomorphic function
$$
f : \frac{\mathbb{C}}{\Lambda} \mapsto \frac{\mathbb{C}}{\Lambda}
$$
where $\Lambda$ is a lattice.
Is it always possible to find another function $\psi : ...
5
votes
4answers
185 views
upper bound on rank of elliptic curve $y^{2} =x^{3} + Ax^{2} +Bx$
I was told the following "Theorem": Let $y^{2} =x^{3} + Ax^{2} +Bx$ be a nonsingular cubic curve with $A,B \in \mathbb{Z}$. Then the rank $r$ of this curve satisfies
$r \leq \nu (A^{2} -4B) +\nu(B) ...
4
votes
3answers
441 views
Integral points on an elliptic curve
Let's start with an elliptic curve in the form
$$E : y^2 = x^3 + Ax + B, \qquad A, B \in \mathbb{Z}.$$
I am wondering about integral points. I know that Siegel proved that $E$ has only finitely many ...
8
votes
2answers
441 views
The modular curve X(N)
I have a question about the modular curve X(N), which classifies elliptic curves with full level N structure. (A level N structure of an elliptic curve E is an isomorphism
from $Z/NZ \times Z/NZ$ to ...
8
votes
2answers
726 views
Definition of the j-invariant of an elliptic curve
It seems that most introductory books on elliptic curves simply state the definition of the j-invariant of an elliptic curve without giving any background on how that definition was conceived. Of ...
4
votes
2answers
413 views
What is a primitive point on an elliptic curve?
While working with elliptic curves for cryptography reasons, I found the notion of a primitive point, but no definition.
For example, $P(0,6)$ is a primitive point on the elliptic curve $y^2\equiv ...
1
vote
1answer
156 views
Algebraic points on an elliptic curve
There is a book about rational points on elliptic curves. What about algebraic points?
2
votes
1answer
98 views
Extracting the value of $y$ from $x$ in an elliptic curve over a finite field
Given an elliptic curve $y^2 = x^3 + ax + b$ over a finite field $\mathbf{F}_p$, how can I retrieve the value of $y$ given the value of $x$?
My knowledge in this area is quite limited, so I ...
13
votes
2answers
1k views
Elliptic Curves and Points at Infinity
My undergraduate number theory class decided to dip into a bit of algebraic geometry to finish up the semester. I'm having trouble understanding this bit of information that the instructor presented ...
1
vote
2answers
121 views
Understanding how to calculate $E_\text{tors}$ of an elliptic curve
In a set of lecture notes, there is an example of calculating the group $E_\text{tors}$ of an elliptic curve. This is the example:
Let $E$ be the elliptic curve
$$y^2=x^3-5x+4.$$
The curve ...
2
votes
2answers
1k views
How elliptic arc can be represented by cubic Bézier curve?
If I have an arc (which comes as part of an ellipse), can I represent it (or at least closely approximate) by cubic Bézier curve? And if yes, how can I calculate control points for that Bézier curve?
3
votes
1answer
160 views
What else can the elliptic integral count?
I just read this document - Jacobi's Four Square Theorem. It shows how to count the number of representations of a number as the sum of four squares.
I can follow the proof but currently it just ...
2
votes
3answers
900 views
Group Law for an Elliptic curve
I was reading this book "Rational points on Elliptic curves" by J.Silverman, and J.Tate, 2 prominent figures in Number theory and was very intrigued after reading the first couple of pages.
The ...
3
votes
3answers
3k views
How could I calculate the rank of this elliptic curve?
The birational change of variables $(u,v) = (\frac{36+y}{6x},\frac{36-y}{6x})$ maps $u^3+v^3=1$ to $y^2 = x^3 - 432$ which has discriminant $-2^{12}\cdot 3^9$.
Using pari/gp we can compute the ...
7
votes
2answers
485 views
A question on FLT and Taniyama Shimura
Sometime back i watched the documentary of Andrew Wiles proving the Fermat's Last theorem. A truly inspiring video and i still watch it whenever i am in a depressed mood. There are certain ...
2
votes
1answer
243 views
Deriving Eulers Addition Theorem for Elliptic Integrals
In the book Elliptic Curves - McKean & Moll we are given the outline for a proof of Eulers addition theorem:
The (projective) quartic $\mathbf y^2 = (1-\mathbf x^2)(1-k^2 \mathbf x^2)$ has ...
2
votes
2answers
218 views
Trying to piece together an integral addition theorem
If we have a curve $C:\{ P(x,y) = 0 \}$ and define $\omega=\frac{\mathrm{d}x}{y}$ then is
$$\int_0^A \omega + \int_0^B \omega = \int_0^{A \oplus B} \omega$$
(with $\oplus$ being addition on a group ...
23
votes
3answers
604 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...
