Questions tagged [elliptic-curves]

Elliptic curves are objects of algebraic geometry met in somewhat advanced parts of number theory. They also appear in applications to cryptography. Use the tag, if this applies. Questions on ellipses should be tagged [conic-sections] instead.

An elliptic curve is a smooth, nonsingular projective curve of genus 1 with a specified point $\mathcal{O}$. It forms an abelian group under point addition. It is an object of much study in number theory, for example in cryptography and integer factorization.

More informally, elliptic curves are cubic curves that have a shape that looks like a donut. They can be defined over any field $K$.

An elliptic curve can be defined by an equation of the form: $$E:y^2=x^3+ax+b$$ with the discriminant $\triangle_E=16(4a^3+27b^2)\ne 0$ so the curve is nonsingular, i.e. its graph has no cusps or intersections.

The elliptic curves with $a=0$ are .

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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 conjecture. It relates the value at $1$ of the derivative…
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Let $k$ be a natural number . Then $3k+1$ , $4k+1$ and $6k+1$ cannot all be square numbers.

Let $k$ be a natural number. Then $3k+1$ , $4k+1$ and $6k+1$ cannot all be square numbers. I tried to prove this by supposing one of them is a square number and by substituting the corresponding $k$ value. But I failed to prove it. If we ignore…
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Rational solutions to $a+b+c=abc=6$

The following appeared in the problems section of the March 2015 issue of the American Mathematical Monthly. Show that there are infinitely many rational triples $(a, b, c)$ such that $a + b + c = abc = 6$. For example, here are two solutions…
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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 things) that the group of units $\mathcal…
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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 problem be solved using pre-university…
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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 manipulation, can be turned into finding…
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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...
bdonlan
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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 there $\nexists \ a,b,c$ such that $I\in \mathbb Z$:…
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How does Wiles' proof fail at $n=2$?

The content is miles outside what I know about. So the question is a mixture of idle curiosity and maybe having this answered somewhere on the Internet. It is likely I will not be able to understand the answer. How exactly does Wiles' proof of…
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Explicit Derivation of Weierstrass Normal Form for Cubic Curve

In page 22-23 of Rational Points on Elliptic Curves by Silverman and Tate, authors explain why is it possible to put every cubic curve into Weierstrass Normal Form. Here are relevant pages: (My question is at the end; I have put a red line across…
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On the integral $\int_0^1\frac{dx}{\sqrt[4]x\ \sqrt{1-x}\ \sqrt[4]{1-x\,\gamma^2}}=\frac{1}{N}\,\frac{2\pi}{\sqrt{2\gamma}}$

V. Reshetnikov gave the interesting integral, $$\int_0^1\frac{\mathrm dx}{\sqrt[4]x\ \sqrt{1-x}\ \sqrt[4]{2-x\,\sqrt3}}=\frac{2\,\sqrt2}{3\,\sqrt[8]3}\pi\tag1$$ After some experimentation, it turns out that more generally, given some…
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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 in his notes. Here it is in paraphrase (assume we…
crasic
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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 transcendental, there is a transcendental $u$ and…
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How are the Tate-Shafarevich group and class group supposed to be cognates?

How can one consider the Tate-Shafarevich group and class group of a field to be analogues? I have heard many authors and even many expository papers saying so, class group as far as I know is the measure of failure of unique factorization of…
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Recent developments in the proof of fermat's last theorem

It's been 20 years since fermat's last theorem was proved by Andrew Wiles. Has there been any simplification in proof in the last 20 years? What I do only know is that different proofs of faltings's theorem were given by Vojta and Bombieri.
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