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 ...
Noam D. Elkies's user avatar
27 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 ...
Will Sawin's user avatar
  • 137k
20 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 ...
Joe Silverman's user avatar
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 ...
Joe Silverman's user avatar
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 "...
Kevin Buzzard's user avatar
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 ...
Zev Klagsbrun's user avatar
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 ...
Myshkin's user avatar
  • 17.4k
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 ...
Pete L. Clark's user avatar
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 ...
Michael Stoll's user avatar
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 ...
Felipe Voloch's user avatar
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 ...
Wojowu's user avatar
  • 27.4k
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 ...
15 votes

Does this conic have a rational point?

The answer is no. If there was, we could assume that $X,Y,Z$ are in $\mathbb Q[u,v]$ and are coprime (since that ring is a UFD). Setting $v=0$ we get $X(u,0)^2+uY(u,0)^2=0$ in $\mathbb Q[u]$, which (...
Wojowu's user avatar
  • 27.4k
13 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})...
Daniel Loughran's user avatar
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} \...
Michael Stoll's user avatar
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 ...
François Brunault's user avatar
11 votes

If $X$ is a genus $g\geq 2$ curve over a number field $K$, then is there a bound on $X_L(L)$ for $L/K$ that depends only on $X$ and $[L:K]$?

Assuming the conjecture that varieties of general type cannot have a Zariski dense set of points over a number field (which is a consequence of the more precise Vojta conjectures, for example), ...
Joe Silverman's user avatar
11 votes
Accepted

Rational points on varieties over local fields

Let us assume that $X$ is smooth and projective for simplicity, given by a number of polynomial equations with coefficients in the ring of integers $\mathcal O$ of $k$. Let $\kappa$ denote the residue ...
Michael Stoll's user avatar
10 votes
Accepted

Does Chabauty-Coleman method give an algorithm for finding rational points?

Conjecturally, yes. Check out Section 4.4 of this paper by Nils Bruin and myself. The point is to combine Chabauty-Coleman with the "Mordell-Weil Sieve". In the following, I will assume for ...
Michael Stoll's user avatar
10 votes
Accepted

Singular curves of genus 1

There's no problem over finite fields, but there is a problem over fields that have a nontrivial Brauer class. If you take a genus $0$ curve that's not rational (say a plane quadric), it will always ...
Will Sawin's user avatar
  • 137k
10 votes
Accepted

Field extensions over which algebraic varieties cannot acquire points

Let $K$ be a finite type field extension of $k$ which corresponds to a rational function field of an algebraic variety $V$ for which the rational points are not Zariski dense. Let $U$ be the ...
Will Sawin's user avatar
  • 137k
9 votes

Determining the Mordell-Weil group of a universal elliptic curve

If you want to do this directly, you could "partially specialize" to, say $y^2 = x^3 + Ax + T$ with $A\in\mathbb C$. Then I don't think it's very hard to show, via a standard descent, that as an ...
Joe Silverman's user avatar
9 votes
Accepted

rational points and a local perturbation of an elliptic curve

A counter-example, showing that the answer is "no" for some $(a_0,b_0)$ can be constructed as follows. Take an elliptic surface over $\mathbb{Q}$ with rank $2$. By Silverman's specialisation theorem, ...
Chris Wuthrich's user avatar
9 votes

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

Indeed I was thinking too linearly. If we change standard notation slightly, and call $r_3(n)$ the number of (unordered) representations of $n$ as a sum of three increasing and distinct squares, then ...
Gerhard Paseman's user avatar
9 votes
Accepted

Subfields of Hilbertian fields

The maximal (pro-)solvable extension $L$ of $\mathbb{Q}$ is not Hilbertian, but every proper finite extension of $L$ is. See Fried-Jarden (third edition, 2008), Example 13.9.5.
Laurent Moret-Bailly's user avatar
9 votes

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

The problem of this question is qualitatively and quantitatively different in some ways from that considered by Andrew Bremner and myself. If we take the cubic, with $N$ as a fixed constant, it is ...
Allan MacLeod's user avatar
9 votes
Accepted

A generator needed for a Z/6 elliptic curve

In this case, it pays to work on the curve $E'$ that is 2-isogenous to $E$, which is given by the equation $$ y^2 = x^3 + 404100192598226941365253x^2+ 1470175712258164849983363482095324897635296971x. $...
Zev Klagsbrun's user avatar
9 votes

Smooth surfaces in positive characteristic

For infinite fields of characteristic $p$, the Zariski open definition makes pretty much as much sense as it does in characteristic zero. Often, but not always, the same argument works. Over finite ...
Will Sawin's user avatar
  • 137k
8 votes
Accepted

Rational points on open subsets of affine space

Here is a short proof that, for an infinite field $k$, and all non-zero polynomials $F \in k[x_1,\ldots,x_n]$ in $n$ variables, there exists an $n$-tuple $a_1,\ldots,a_n \in k$ such that $$ F(a_1,\...
R.P.'s user avatar
  • 4,745
8 votes

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

To add on to Timo's suggestion, Hindry and Silverman have a book on Diophantine geometry which gives a self-contained proof of Faltings' Theorem. However the proof covered is not the original proof ...
ABarrios's user avatar
  • 163

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