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Questions tagged [rational-points]

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How many points are there on an elliptic curve reduced at a bad prime?

Given an elliptic curve $E$ defined over $\mathbb{Z}$, and a prime $p$, I know that Hasse's theorem gives, when $p$ is a good prime, a relation between the number of solutions over $\mathbb{F}_{p^n}$ ...
Anonymous's user avatar
20 votes
4 answers
3k views

Does $(x^2 - 1)(y^2 - 1) = c z^4$ have a rational point, with z non-zero, for any given rational c?

I need this result for something else. It seems fairly hard, but I may be missing something obvious. Just one non-trivial solution for any given $c$ would be fine (for my application).
John R Ramsden's user avatar
7 votes
1 answer
334 views

Rational points on smooth compactifications

Let $X$ be as smooth variety over a field $k$ of characteristic $0$. Consider the following statements: The variety $X$ has no $k((t))$-rational points. No smooth compactification of $X$ has a $k$-...
Wanderer's user avatar
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8 votes
3 answers
807 views

Smart elliptic curve rational point search given Reg*#Sha

Hi folks, Let E be a global minimal model of an elliptic curve over QQ, with a 2-torsion point which generates the torsion subgroup, and with Mordell-Weil rank 1 (under BSD). Let RegSha be equal to ...
Iftikhar Burhanuddin's user avatar
3 votes
1 answer
483 views

Brauer-Manin obstruction and Hasse principle

I am looking for varieties without $\mathbf{Q}$-rational points where the absence can be explained by the Brauer-Manin obstruction, but not by the absence of adelic points varieties without $\mathbf{...
user avatar
0 votes
0 answers
227 views

Ideal of real points in $\mathbb{C}[x_0,x_1,\dotsc,x_k]$

I fell a little uncomfortable with this real stuff. The question here is more general, but we can suppose $\mathbb{K}=\mathbb{R}$. Take a set of (distinct) points in $\mathbb{P}^n$, the complex ...
pirignao's user avatar
5 votes
1 answer
316 views

Existence of a non-trivial zero (in the rational cyclotomic field) of a form

It is well known that if a field K is quasi-algebraically closed (i.e. all forms with coefficients in K of degree d in n > d variables have a non-trivial zero in K) then it has no central divison ...
Alessandro Macedo's user avatar
-1 votes
1 answer
802 views

Genus of algebraic curves with unknown degree

I am not sure if this is a valid question but posting any way: Say I am over $\mathbb{F}_{p}$ for a prime $p$. I have a curve of form $x^{2} = f(y)$ where $f(y)$ has an unknown form (and hence ...
17 votes
1 answer
1k views

Totally rational polytopes

Define a convex polytope in $\mathbb{R}^d$ as totally rational (my terminology) if its vertex coordinates are rational, its edge lengths are rational, its two-dimensional face areas are rational, etc.,...
Joseph O'Rourke's user avatar
9 votes
2 answers
929 views

Shortest irrational path

What is the shortest curve $\gamma$ in $\mathbb{R}^2$ from the origin $o=(0,0)$ to a rational point $p=(a,b)$ that (a) passes through no other rational point, and (b) contains no point a ...
Joseph O'Rourke's user avatar
6 votes
1 answer
2k views

k rational points and base change

This could be a tricky question but could help me to better understand these very interesting things. Let $X$ be an algebraic variety over a field $k$ (in the sense of a k-scheme like in Qing Liu), $...
Srks's user avatar
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17 votes
1 answer
2k views

Rational points à la Chabauty-Coleman

I have been trying to learn the method of Chabauty and Coleman to find rational points on curves; I have been reading an exposition by McCallum and Poonen which was pointed out to me by Emerton in ...
Barinder Banwait's user avatar
20 votes
3 answers
6k views

Closed vs Rational Points on Schemes

Background: When Ueno builds the fully faithful functor from Var/k to Sch/k he mentions that the variety $V$ can be identified with the rational points of $t(V)$ over $k$. I know how to prove this on ...
Matt's user avatar
  • 970
9 votes
1 answer
399 views

Existence of hyperelliptic curve with specific number of points in a family

Hi, the following question was posed to me, it apparently has applications for linear codes. Let n>1, and $K = \rm{GF}(2^n)$. Let $k$ be coprime to $2^n-1$. Does there always exist $a \neq 0$ in $K$ ...
Dan Petersen's user avatar
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10 votes
4 answers
1k views

Sums of cubes and more

It's well-known that every natural number can be written as a sum of 4 squares of integers. Has there been any recent progress about the similar problem for the cubes, 4-th powers and so on? I ...
Ilya Nikokoshev's user avatar
7 votes
2 answers
509 views

How can you find small denominators inside triangles?

Darsh asked over at the 20 questions seminar: Take a triangle in R^2 with coordinates at rational points. Can we find the smallest denominator point in the interior? (Consider denominator of an ...
Kim Morrison's user avatar
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