All Questions
Tagged with rational-points ag.algebraic-geometry
109 questions
2
votes
2
answers
541
views
A new simple formula is needed
The following question is related to the families of high rank elliptic curves with torsion subgroup $\mathbb{Z}/6\mathbb{Z}$.
The SageMath/Python code below produces a list of small fractions $a$ for ...
- 913
2
votes
1
answer
187
views
Is every sufficiently general monic quartic rational square infinitely often?
Let $f(x)=x^4+b_3 x^3+ b_2 x^2+b_1 x + b_0$.
Let $g(x)=x^4 f(1/x)$. Let $C : g(x)=y^2$.
$C$ is birationally equivalent to $f(x)=y^2$.
The constant coefficient of $g(x)$ is $1$ since $f$ is monic
and $(...
- 149k
2
votes
1
answer
269
views
Perfect square quadratic expression
For a given rational $c\ne-1$, I need to find a rational $x\ne20$ with a small denominator such that $(5cx+100)(5cx-64c+36)$ is a perfect square.
I start with
$y^2=(5cx+100)(5cx-64c+36)$
and ...
- 913
16
votes
0
answers
274
views
Why should an abelian variety with few places of bad reduction and a lot of endomorphisms not have many points?
In the paper "Points of Order 13 on Elliptic Curves" by Mazur-Tate, they say in the introduction:
It seemed ... that if such an abelian variety $J$, which has bad reduction at only one ...
- 7,746
8
votes
0
answers
135
views
Distribution of rational points in the real locus of a planar algebraic curve
Let $C$ be a smooth projective geometrically connected curve over $\mathbb{Q}$. Assume that $g(C)=3$ and that $C$ is not hyperelliptic. Then the canonical sheaf defines a closed immersion $C\to\mathbb{...
CommunityBot
- 1
3
votes
1
answer
395
views
Finding $K$-rational points on $X_0(35)$
Let $K=\mathbb{Q}(\sqrt{-2})$. How can I compute the $K$-rational points on the modular curve $X_0(35)$?
Recall that $X_0(35)$ is a hyperelliptic curve of genus 3 and has the simplified affine model:
\...
- 11.3k
4
votes
2
answers
793
views
Pointless, non-singular, absolutely irreducible affine plane curves over finite fields
We think the following is true:
For all sufficiently large primes $p$ and all natural $g \ge 1$, there
exists affine plane curve $f(x,y)=0$ over $\mathbb{F}_p$ which
is non-singular, absolutely ...
- 2,051
4
votes
1
answer
423
views
A generator needed for a Z/6 elliptic curve
We are searching for rank $8$ elliptic curves with the torsion subgroup $\mathbb{Z}/6$ using newly discovered families similar to Kihara's as described in
A. Dujella, J.C. Peral, P. Tadić, ...
- 913
2
votes
1
answer
600
views
Density of rational points over finite fields, an estimate of Lang-Weil constant
Let $X\hookrightarrow\mathbb P^n_{\mathbb F_q}$ be a geometrically integral hypersurface over the finite field $\mathbb F_q$ of degree $\delta$. In order to estimate the number of its rational points, ...
- 403
8
votes
1
answer
904
views
Hard: One more generator needed for a Z/6 elliptic curve
We are searching for rank 8 elliptic curves with the torsion subgroup $\mathbb{Z}/6$ using newly discovered families similar to Kihara's as described in
A. Dujella, J.C. Peral, P. Tadić, Elliptic ...
- 4,713
7
votes
1
answer
218
views
Subfields of Hilbertian fields
This question is about the Remark on the top of page 22 of Serre's Topics in Galois Theory, available here :
http://www.ms.uky.edu/~sohum/ma561/notes/workspace/books/serre_galois_theory.pdf
My ...
- 19.6k
3
votes
0
answers
339
views
Integral points on affine varieties
Consider Siegel's theorem. It says that for a smooth affine algebraic curve $C$ over $\mathbb{Q}$ such that $g(C)>0$ any model $\mathcal{C}$ of $C$ over $\mathbb{Z}$ has finitely many $\mathbb{Z}$-...
CommunityBot
- 1
3
votes
1
answer
190
views
Connected sum of algebraic curves, handlebody decomposition, and induction on genus
Are there any nice methods of taking algebraic curves $C_1, C_2$ of genera $g_1,g_2$ and producing a curve $C_3 = C_1 \# C_2$ of genus $g_3 = g_1+g_2$? I'm imagining doing this over any field, but ...
- 2,384
0
votes
0
answers
279
views
Computing the genus of a plane curve
Let $b(x)=x^4 + 3x^3 + 3x^2 + 2x + 1$, and let $a(x)\in \mathbb Z[x]$ be a separable polynomial. Let $C$ be the plane curve defined by $(y^2+(x+x^2+x^3)a(x))^2-a(x)^2b(x)=0$. I would need to show that ...
- 101
10
votes
1
answer
562
views
Are there infinitely many real multiplication fields of abelian surfaces over $\mathbb Q$?
Do there exist infinitely many real quadratic fields $F$ such that there is an abelian surface $A$ over $\mathbb Q$ whose ring of endomorphisms, tensored with $\mathbb Q$, is $F$?
Do there exist ...
CommunityBot
- 1
4
votes
1
answer
328
views
Submersion implies many rational points in image?
Let $A \colon V \to W$ be a surjective linear map
(defined over $\mathbb{Z}$),
inducing a projection
$\alpha \colon \mathbb{P}(V) \to \mathbb{P}(W)$.
Let $X \subseteq \mathbb{P}(V)$ and $Y \...
CommunityBot
- 1
6
votes
1
answer
366
views
Breaking a morphism with generic fiber $\mathbb{F}_n$
Assume we are working over $\mathbb{C}$, and we have a projective morphism with connected fibers $f: X \rightarrow Z$ whose geometric generic fiber $X_\overline{\eta}$ is isomorphic to a Hirzebruch ...
- 7,722
9
votes
2
answers
792
views
Rational points techniques on curves not using their Jacobian
Let $C/K$ be a curve of genus > 2 over a number field $K$ and suppose there exists a $p \in C(K)$. Then a recurring theme in studying $C(K)$ is using the map $C \to J(C)$ normalized by sending $p$ to ...
- 11.3k
20
votes
5
answers
2k
views
Rational solutions of the Fermat equation $X^n+Y^n+Z^n=1$
As a generalisation to the equation of Fermat, one can ask for rational solutions of $X^n+Y^n+Z^n=1$ (or almost equivalently integer solutions of $X^n+Y^n+Z^n=T^n$).
Contrary to the case of Fermat, ...
- 2,811
2
votes
0
answers
310
views
Rational point on variety over function field
This is one of the theorems in Field Arithmetic written by M. Fried and M. Jarden as following which is Proposition 13.4.6 in that monograph:
Every field K has a regular extension F which is PAC ...
- 151
7
votes
1
answer
557
views
Does Chabauty-Coleman method give an algorithm for finding rational points?
Let $X$ be a curve of genus $g\geq 2$ over a number field $K$. If $\mathrm{rk} \,\mathrm{Jac}\, X$ is less than $g$ there is a $p$-adic method of bounding $\# X(K)$ due to Chabauty and Coleman (see ...
- 6,051
4
votes
0
answers
313
views
Action of the Picard Scheme of an Elliptic Fibration
Suppose that we have a surface $X$ defined over a field $k$ (I am interested in $k$ being a number field) and an elliptic fibration $f: X \rightarrow \mathbb{P}^1$, i.e. $f$ is proper and almost all ...
- 241
8
votes
1
answer
986
views
Why study unirational and rational varieties?
I am new to the study of unirational and rational varieties, but I want to know the motivation for why mathematicians started to study these conditions. The reasons that I could list to study ...
- 39.3k
2
votes
0
answers
376
views
Number of rational points of a singular cubic surface over a finite field
I have a (geometrically irreducible) cubic surface defined over a finite field $F_q$ with three non-$F_q$-rational singularities (defined over the cubic extension of $F_q$).
Counting the number of $...
- 21
10
votes
4
answers
1k
views
Possible groups of K-rational points for elliptic curves over arbitrary fields
It is known that the group $C(\Bbb R)$ has at most two connected components, and the connected component of the identity is isomorphic to $U(1)$ as a topological group (trivially) and $C(\Bbb Q)$ is ...
- 17.6k
2
votes
1
answer
657
views
Rational points on open subsets of affine space [closed]
Let $k$ be an infinite field. Assume that the index of the algebraic closure $\bar{k}$ over $k$ is strictly greater than $2$. Let $U$ be a non-empty open subset of some affine space over $k$. Is it ...
- 4,745
1
vote
0
answers
122
views
Existence of rational points on the image of a proper morphism
Let $K/F$ be a field extension, and let $X$ and $Y$ be affine varieties over $F$. (E.g. they are defined by polynomials over $F$.) Suppose $X$ contains $F$-points. Now view $X$ and $Y$ as $K$-...
- 565
5
votes
1
answer
825
views
Understanding Siegel's Theorem on integral points
Siegel's theorem states the following:
Let $C$ be a smooth projective curve over a number field $K$. Let $\tilde C\subset C$ be an open affine subvariety, and $i:\tilde C\hookrightarrow \mathbb{A}^...
- 30.5k
6
votes
1
answer
185
views
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 $[L:K]$?
If $X$ is a genus $g\geq 2$ curve over a number field $K$, then $X(K)$ is finite by Falting's Theorem. My question is how does $X_L(L)$ behave for finite field extensions $L/K$? In particular, is ...
- 47.4k
7
votes
1
answer
508
views
What is the exact statement about uniform boundedness of rational points on curves of genus greater than one? Singular points can be unbounded
According to several sources, it is conjectured (or at least believed)
that the rational points of curves over the rationals of genus $g > 1$
are uniformly bounded by $g$. E.g. here p. 1.
Assuming ...
- 17.6k
11
votes
5
answers
4k
views
How much do I need to learn algebraic geometry to understand arithmetics over number fields
I am at the stage of learning. Mostly, I am attracted by algebraic number theory. Roughly speaking, I am interested in the rational points of algebraic varieties. I am little bit afraid to start to ...
- 1,499
2
votes
2
answers
440
views
Rational points on towers of curves
Let $\ldots \to X_n \to X_{n-1} \to \ldots \to X_0$ be etale maps between smooth projective curves of genera $g(X_n)>1$, all defined over a fixed number field $K$. By Faltings' Theorem, we know ...
- 10.7k
5
votes
0
answers
299
views
A relative version of Hensel's lemma?
Let $k$ be a $p$-adic field with integer ring $\mathcal{O}_k \subseteq k$, maximal ideal $m_k \subseteq \mathcal{O}_k$ and residue field $\mathbb{F}_q = \mathcal{O}_k/m_k$. Let $X$ be a smooth, ...
- 9,491
-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 ...
CommunityBot
- 1
1
vote
2
answers
203
views
Counting number of $2\times 2$ unimodular matrices of particular type
From set of numbers from $\Bbb S=\{0,1,\dots,m\}$, how many distinct $3\times 3$ unimodular matrices parametrized by $(a,b,c,d,e,f)\in\Bbb S^6$ of following type can one form?
\begin{bmatrix}
a^2 &...
CommunityBot
- 1
3
votes
0
answers
559
views
Rank of the Jacobian of twists of hyperelliptic curves
Suppose that a hyperelliptic curve $C$ of genus $g \geq 4$ is given by the equation
$$\displaystyle C: y^2 = a_0 x^{2g+2} + a_1 x^{2g+1} + \cdots + a_{2g+2} = f(x).$$
The Jacobian variety $J(C)$ of ...
- 11.3k
4
votes
2
answers
411
views
Find all possible rational values of a parametric quartic such that it is reducible
Description: Given the following parametric quartic polynomial
$y^4 - 28 z y^3 - 14 (656 - 328 z + 83 z^2) y^2 +
4 z (-20464 + 10232 z + 3409 z^2) y +
91 (62208 - 62208 z + 41504 z^2 - 12976 z^3 +...
- 11.3k
3
votes
2
answers
520
views
cohomological obstructions and rational points
Let $X$ a (nice) scheme over $\mathbb{Q}$. Are there cohomolgical obstructions answering the following questions:
1) is $X(\mathbb{Q})$ an empty set ?
2) is $X(\mathbb{Q})$ a finite (non empty) set ...
- 17.4k
14
votes
1
answer
1k
views
Elliptic curves and connected components
Are there elliptic curves of positive rank with two real connected components
in which all the rational points lie only on one component?
Concrete examples are really appreciated.
CommunityBot
- 1
1
vote
0
answers
211
views
Coarse moduli spaces and rational points [closed]
Let $K$ be a field (not necessarily algraically closed). Let $\mathcal{F}$ be a contravariant functor from the category of schemes over $K$ to sets and $M$ be a coase moduli space for the functor. So, ...
- 2,323
12
votes
1
answer
617
views
What is our current knowledge on the structure of J_0(N)(Q) and J_1(N)(Q)
The question in the title naturally breaks up in two parts, namely the torsion part and the rank part. I already read about some results on both the torsion and the rank part. And I want to know ...
- 20.4k
12
votes
2
answers
424
views
Existence of local sections
I would like to know when the property of "having a section" for a morphism of varieties in characteristic $0$ can be detected by spreading out to characteristic $p$.
Take a number field $K$, and let ...
- 4,111
1
vote
0
answers
193
views
Existence of a curve with no points over finite separable field extensions
Does there exist a field $K$, and a smooth projective geometrically connected curve $C$ over $K$ such that, for all finite separable field extensions $L/K$ the curve $C$ has no $L$-rational points?
I ...
- 51
1
vote
1
answer
543
views
Infinite residue field extensions and algebraic closure of residue fields
Let $X$ be a $K$-scheme of finite type over a field $K$, let $L$ be an extension field of $K$, let $X_L := L \times_K X$, and let $p:X_L \rightarrow X$ be the projection. For each $x \in X_L$ we get ...
- 10.7k
4
votes
1
answer
874
views
Closed points of field extension of k-scheme under projection
I really couldn't figure out the answer to the following question: Let $X$ be a scheme of finite type over a field $k$ and let $K$ be an extension field of $k$. Let $X_K := K \times_k X$ be the base ...
- 6,543
0
votes
2
answers
400
views
"rationality" of divisors
Let $X$ be a smooth projective variety over some field $k$. Then each closed point $x$ has an associated residue field $k(x)$ which is a finite extension of $k$ and a point is rational when $k(x)=k$.
...
- 47.4k
11
votes
2
answers
791
views
Geometrically unirational varieties that are not unirational
By a variety over a field $k$, I mean a scheme that is separated and
of finite type over $k$. I indicate changes of the base ring by
subscripts.
Does there exist a smooth and projective variety $V$ ...
- 4,111
0
votes
0
answers
214
views
Deformation of rational points in a family
Let $\mathcal{X} \to B$ be a family of smooth projective varieties over a field $K$ (possibly finite). Assume that each fiber $\mathcal{X}_b$ of the family has a $K$-rational point. Fix a pair $(p,\...
- 2,032
1
vote
1
answer
241
views
Tricks to produce examples of hypersurfaces with index greater than $1$
Recall, index of an algebraic scheme $X$ is defined to be the greatest common divisor of the degrees of the space of zero cycles on $X$. I am interested in examples of hypersurfaces in $\mathbb{P}^n_K$...
- 6,262
6
votes
1
answer
489
views
Simple field extension and rational points
Let $F$ be an infinite field and $f$ a homogeneous form on $F$ such that $f$ has no non-trivial zero in $F$. Let $F'$ be a finite extension of $F$ such that $f$ has a non-trivial zero in $F'$. Is it ...
- 22.5k