# Tagged Questions

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

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### Gross-Zagier formulae outside of number theory

(Edit: I have asked this question on MO.) The Gross-Zagier formula and various variations of it form the starting point in most of the existing results towards the Birch and Swinnerton-Dyer ...
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### The resemblance between Mordell's theorem and Dirichlet's unit theorem

The first one states that if $E/\mathbf Q$ is an elliptic curve, then $E(\mathbf Q)$ is a finitely generated abelian group. If $K/\mathbf Q$ is a number field, Dirichlet's theorem says (among other ...
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### 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...
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### How to compute rational or integer points on elliptic curves

This is an attempt to get someone to write a canonical answer, as discussed in this meta thread. We often have people come to us asking for solutions to a diophantine equation which, after some clever ...
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### 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 ...
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### Find integer in the form: $\frac{a}{b+c} + \frac{b}{c+a} + \frac{c}{a+b}$

Let $a,b,c \in \mathbb N$ find integer in the form: $$I=\frac{a}{b+c} + \frac{b}{c+a} + \frac{c} {a+b}$$ Using Nesbitt's inequality: $I \ge \frac 32$ I am trying to prove $I \le 2$ to implies ...
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### 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 ...
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### More elliptic curves for $x^4+y^4+z^4 = 1$?

(Note: This has been updated to be similar with this MO post.) There are exactly 22 known primitive solutions to, $$a^4+b^4+c^4 = d^4\tag{1}$$ with $d<10^{11}$. Noam Elkies showed that $(1)$ as, ...
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### History of elliptic curves

In one sense elliptic curves are a rather modern object as some of its properties have been studied only in the last century or so. But in another sense there are a very classical object for studying ...
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### Reference request in number theory for an analyst.

I am a confirmed mathochist. My background is in analysis, and fairly traditional analysis at that; mainly harmonic functions, subharmonic functions and boundary behaviour of functions, but I have for ...
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### The term “elliptic”

There are many things which are called “elliptic” in various branches of mathematics: Elliptic curves Elliptic functions Elliptic geometry Elliptic hyperboloid Elliptic integral Elliptic modulus ...
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### 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 ...
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### 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 ...
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### why say complex multiplication of elliptic curves is beautiful

David Hilbert is said to have remarked that the theory of complex multiplication of elliptic curves was not only the most beautiful part of mathematics but of all science. Just as the title asked. ...
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### 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 ...
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### Clarifying a comment of Serre

Let $\rho_{\ell}$ be the "mod $\ell$" Galois representation associated to an elliptic curve $E/K$ (i.e., corresponding to the action of Galois on the $\ell$-torsion points). Serre proved that in the ...
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### Find $x\in \mathbb{Z}$ such that $54x^3+1$ is a cube

Find $x\in \mathbb{Z}$ such that $54x^3+1$ is a cube. I found $x=0$, any others ?
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### Diophantine equation $x^2 + xy + y^2 = \left({{x+y}\over{3}} + 1\right)^3$.

Solve in integers the equation$$x^2 + xy + y^2 = \left({{x+y}\over3} + 1\right)^3.$$
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### Cube root of discriminant of elliptic curve

Let $E/K$ be an elliptic curve over a field $K$, with discriminant $\Delta$. Then the polynomial $x^3-\Delta$ has a root (and hence all roots since Galois) in $K(E[3])$; this can be shown laboriously ...
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### Integer solutions for $x^3+2=y^2$?
I've heard a famous result that $26$ is the only integer, such that $26-1=25$ is a square number and $26+1=27$ is a cubic number.In other words, $(x,y)=(5,3)$ is the only solution for $x^2+2=y^3$. ...