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

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4
votes
1answer
179 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
281 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
1k 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
804 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 ...
10
votes
2answers
1k 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
921 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
191 views

Algebraic points on an elliptic curve

There is a book about rational points on elliptic curves. What about algebraic points?
2
votes
1answer
119 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 ...
17
votes
2answers
2k 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
159 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 ...
3
votes
2answers
2k 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
165 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 ...
4
votes
3answers
1k 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 ...
4
votes
3answers
4k 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
522 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
262 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
230 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 ...
25
votes
3answers
928 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...