124 votes
Accepted

Estimating the size of solutions of a diophantine equation

This problem turned out to be much more interesting than I originally thought. Let me give my solution, which seems to be slightly different from (but essentially the same as) the solution in the ...
75 votes

Estimating the size of solutions of a diophantine equation

This exact problem is the subject of the paper "An Unusual Cubic Representation Problem" by Andrew Bremner (ASU) and myself. It was published in Volume 43 (2014) of Annales Mathematicae et ...
41 votes
Accepted

Rational points on the "quintic circle" $x^5 + y^5 = 7$

There is an action of $\mu_5$, the group of fifth roots of unity, on your curve, given by $\zeta \cdot (x,y) = (\zeta x, \zeta^{-1} y)$. The quotient by this group action is the hyperelliptic curve $$...
33 votes
Accepted

Non-trivial solutions for $-(c^2-d^2)(a^2-b^2)=2(ad-bc)(bd+ac)$?

There is no such solution. Let $$ Q(a,b,c,d) = 2(ad-bc)(bd+ac) + (a^2-b^2)(c^2-d^2) $$ be the difference between the two sides of the equation, so we seek to solve $Q(a,b,c,d) = 0$. This is a ...
26 votes
Accepted

Why is there a $\sqrt{5}$ in Hurwitz's Theorem?

As Terry mentions in the comments, the reason for the $\sqrt{5}$ is that the limiting case, the golden ratio, forces it. There is a very neat explanation of all of this in the classic number theory ...
  • 17.1k
25 votes
Accepted

How much do I need to learn algebraic geometry to understand arithmetics over number fields

Well if you want to count rational points on varieties than you probably want to know what abelian varieties are, and general type varieties, and Fano varieties, and K3 surfaces, and what Azumaya ...
  • 119k
22 votes
Accepted

Rational solutions of the Fermat equation $X^n+Y^n+Z^n=1$

It has been conjectured by Euler that this equation has no solutions in positive integers when $n\geq 4$. When $n=4$, this was disproved by Elkies in the paper [Elkies, On A4+B4+C4=D4] in a very ...
19 votes
Accepted

Elliptic curves and connected components

Yes. It is not hard to find an example: Take $$E \colon y^2 = x^3 - 12 x - 1\,.$$ Then $E(\mathbb Q) \cong \mathbb Z$ and $P = (5, 8)$ is a generator (according to Magma). Since $P$ is on the ...
19 votes

Galois Representations and Rational Points

In general one can say very little. There are some positive results (as indicated in the comments) in special cases, but the below example kills any hope that one can say something in general. NB "...
19 votes

Possible groups of K-rational points for elliptic curves over arbitrary fields

The structure of $E(K)$ for $K$ a complete local field, say a finite extension of $\mathbb Q_p$ or $\mathbb C_p$, is quite standard. Let $E_0(K)$ denote the set of points with good reduction. Then ...
19 votes

What is the smallest sphere whose surface includes 100 integer points?

Here are a few facts about this problem, quoting mostly from Local statistics of lattice points on the sphere by Jean Bourgain, Peter Sarnak, Zeév Rudnick: ''A celebrated result of Legendre/Gauss ...
18 votes
Accepted

rational points of a hyperelliptic curve

By now there is a fairly rich literature on computing the set of rational points on curves of higher genus, see for example my survey paper on "Rational points on curves". What one can do for your ...
17 votes
Accepted

Hard: One more generator needed for a Z/6 elliptic curve

A set of points that generate $E(\mathbb{Q})$ modulo torsion is given by ...
16 votes
Accepted

Possible groups of K-rational points for elliptic curves over arbitrary fields

By Mordell-Weil, for any number field $K$ we have $$C(K)=\mathbb{Z}^r \times E(K)_{\mathrm{tors}}$$ As you mention, Mazur showed all the possible options for $E(\mathbb{Q})_{\mathrm{tors}}$ in his ...
  • 17.1k
16 votes

Possible groups of K-rational points for elliptic curves over arbitrary fields

The answer to one possible interpretation of the title question -- vary over all elliptic curves over all fields and ask which groups arise -- is given in this paper. With regard to the structure of ...
16 votes

rational points of a hyperelliptic curve of genus 3

It turns out that $C(K) = C(\mathbb Q) = \{\infty_+, \infty_-, (0,1), (0,-1), (1,1), (1,-1)\}$. To see this, consider a point $P \in C(K)$ and write $\bar{P}$ for its image under the nontrivial ...
15 votes
Accepted

Determining the Mordell-Weil group of a universal elliptic curve

Specialize $a,b$ to functions giving the universal elliptic curve over the modular curve $X_0(N)$. These are known to have rank zero over the function field of the modular curve with coefficients over ...
15 votes
Accepted

Around the diophantine equation $\frac{a}{2b+3c}+\frac{b}{2c+3a}+\frac{c}{2a+3b}=\text{odd integer}$, over positive integers

The conjecture is false, and in fact there exists a positive integer solution for $$\frac{a}{2b+3c}+\frac{b}{2c+3a}+\frac{c}{2a+3b}=5,$$ though I was unable to find it explicitly. I will explain how ...
  • 24.3k
15 votes
Accepted

Status of $x^3+y^3+z^3=6xyz$

(Collecting comments into a community wiki answer.) There is a standard method for transforming a smooth cubic into Weierstrass form. See for example Section 1.3 or Appendix B of Silverman and Tate's ...
14 votes

Rational points on the "quintic circle" $x^5 + y^5 = 7$

In his 1825 paper, Lejeune Dirichlet proved that the equation $x^5 + y^5 = cz^5$ has no nontrivial solution with x and y coprime integers and z integer for a rather large class of integers c. His ...
  • 1,205
14 votes
Accepted

Average height of rational points on a curve

First, may I change your notation a bit? Usually one uses $H(p/q)=\max\{|p|,|q|\}$ for the (multiplicative) height of a rational number, and $h(p/q)=\log H(p/q)$ is the logarithmic height. So I'll use ...
13 votes

what is the maximum number of rational points of a curve of genus 2 over the rationals

Here is some more information. The curve that establishes the current record is obtained from a K3 surface $S$ that was found by Noam Elkies. $S$ is a double cover of ${\mathbb P}^2$ ramified above a ...
13 votes
Accepted

Find all rational solutions of this diophantine-equation?

The number of rational solutions to your equation is finite. In short: your equation defines a genus $3$ curve, as follows from a straightforward computation and an application of Riemann--Hurwitz; ...
  • 4,430
13 votes

Conics, rational points and probability

Problems of this type are considered by Serre in the paper: Serre - Spécialisation des éléments de $\mathrm{Br}_2(\mathbb{Q}(T_1,\ldots, T_n))$ The case relevant to you is Exemple 4. Here Serre ...
13 votes

Finding $K$-rational points on $X_0(35)$

The group $J_0(35)(\mathbb Q)$ (where $J_0(35)$ is the Jacobian of $X_0(35)$) has rank 0 (as shown for example by a 2-descent computation in Magma); it is isomorphic to ${\mathbb Z}/24{\mathbb Z} \...
12 votes

Find all rational solutions of this diophantine-equation?

Taking the equation in Joe Silverman's comment as the defining equation and asking Magma: > A := AffineSpace(Rationals(), 2); > C := Curve(A, q^2*p^4 + (-4*q^3+4*q)*p^3 - 2*q^2*p^2 + (4*q^3-4*...
12 votes

Possible groups of K-rational points for elliptic curves over arbitrary fields

I assume in the question that $C = E$ is an elliptic curve. First your claim that $E(\mathbb{R}) = U(1)$ is false; I mean $E(\mathbb{R})$ can be disconnected. The correct result is that $E(\mathbb{R})...
12 votes
Accepted

Are there infinitely many real multiplication fields of abelian surfaces over $\mathbb Q$?

A conjecture of Coleman asserts that only finitely many rings arise as the endomorphism ring of an abelian variety of given dimension defined over a number field of given degree. See [1] for an ...
11 votes

Rational solutions of the Fermat equation $X^n+Y^n+Z^n=1$

Here are some references: MR2077618 Gundersen, Gary G.; Tohge, Kazuya Entire and meromorphic solutions of $f^5+g^5+h^5=1.$ Symposium on Complex Differential and Functional Equations, 57–67, Univ. ...
11 votes
Accepted

Conics, rational points and probability

For any given prime number $p > 2$ the probability that there is no $p$-adic point is $\ge c/p$ for some constant $c > 0$ indpendent of $p$. Since the sum over $1/p$ diverges, this implies that ...

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