Questions tagged [rational-points]
The rational-points tag has no usage guidance.
216 questions
7
votes
0
answers
140
views
Average number of $\mathbb{F}_p$-points over twists of a variety
Let $p \gg 1$ be a sufficiently large prime. I recently stumbled across a fascinating fact about the number of $\mathbb{F}_p$-points on elliptic curves over finite fields. Specifically, we have:
Fact ...
1
vote
0
answers
146
views
Can we find curves with many rational points using linear algebra?
Probably this is impossible, but let us try.
Working over $\mathbb{Q}[x_1,...,x_n]$.
Let $T_i$ be $n$ sets of rationals with cardinality $B$.
Assume we are given $n-2$ linear equations $f_i$ which are ...
- 25.4k
0
votes
0
answers
99
views
Does the smooth locus of any toric variety built from a fan always contain a rational point?
Let $k$ be an arbitrary field and $X$ be a toric variety built from a fan, defined over $k$.
Does the smooth locus of $X$ always contain a $k$-rational point? Why?
- 639
2
votes
2
answers
321
views
The Hasse-Weil inequality
What is the Hasse-Weil inequality (in particular, a lower bound) for singular projective curves over finite fields which are not geometrically irreducible?
- 918
0
votes
1
answer
95
views
Number of points of Fermat surfaces $X^n + Y^n - U^n - V^n = 0$
Let $n$ be a positive integer such that $n^2 + n + 1$ is a prime, and consider the Fermat surface $F$ given by the equation $X^n + Y^n - U^n - V^n = 0$ (where we work with homogeneous coordinates $(x :...
- 4,605
7
votes
1
answer
287
views
What are the intermediate semisimple groups of type A?
Background: The first examples one sees of reductive groups over a field $k$ are $\text{GL}_n$, $\text{SL}_n$, and $\text{PGL}_n$. We all know the definitions of $\text{GL}_n$ and $\text{SL}_n$, and ...
- 835
7
votes
1
answer
399
views
Rational points on an analytic curve
Let $\Gamma$ be a transcendental analytic curve in $\mathbb{R}^2$. I am interested in the topology of its rational points $\Gamma(\mathbb{Q}):=\Gamma\cap\mathbb{Q}^2$.
We know by Pila-Wilkie that if $\...
- 73
0
votes
0
answers
49
views
Is there a correlation between the bifurcation points of dynamical system and the integral points of the elliptic curve $E_d$?
Motivated by an interest in the interplay between dynamical systems and elliptic curves also On a question of Mordell, I derived a dynamical system corresponding to the elliptic curve:
$
E_d: Y^2 = X^...
1
vote
0
answers
122
views
Rational points on an elliptic curve the denominator of x is a square
Let $f \in \mathbb Q[x]$ be a squarefree cubic polynomial with nonzero constant coefficient and consider the elliptic curve $E : y^2 = f(x)$.
Define $E(\mathbb Q)' \subseteq E(\mathbb Q)$ as
$$\left \...
- 2,224
5
votes
1
answer
251
views
Integral points on homogeneous spaces over a DVR
Let $R$ be a DVR (possibly mixed characteristic) with fraction field $K$. Let $V \to \operatorname{Spec} R$ be a smooth affine scheme with a transitive action of $GL_{n,R}$ so that each geometric ...
- 3,290
0
votes
0
answers
108
views
Isogeny classes for elliptic curves over quadratic field
Question. Is it possible for an elliptic curve $E$ over quadratic field $K$ to have two separate (yet connected) isogeny classes?
There are two $\mathbb{Z}/14\mathbb{Z}$ elliptic curves, $E_1$ and $...
- 913
3
votes
0
answers
91
views
Mattuck's Theorem for abelian varieties for a non-locally compact field
Let $A$ be an abelian variety of dimension $d$ defined over a complete ultrametric field $K$ of dimension $0$. Let us put on $A(K)$ the topology induced by the one of $K$ (for example, following ...
- 95
2
votes
0
answers
73
views
Number of rational points of a connected reductive group in a compact subset
Let $G$ be a connected reductive $\mathbb{Q}$-group. Let $\mathbb{A}$ denote the ring of adèles of $\mathbb{Q}$. Let $B \subset G(\mathbb{A})$ be a compact, let $x \in G(\mathbb{A})$ and consider the ...
- 81
2
votes
0
answers
120
views
Upper and lower bounds on the number of solutions to the equation $\frac{\pi}{4} = \sum_{k=1}^{n} c_{k} \arctan \left(\frac{1}{x_{k}} \right) $
Background
The Norwegian mathematician and astronomer Carl Størmer did important work on the equation
$$\frac{\pi}{4} = \sum_{k=1}^{n} c_{k} \arctan \left(\frac{1}{x_{k}}\right), \label{1}\tag{1} $$
...
- 4,829
2
votes
0
answers
136
views
Similar to a $d$-twist but over a cubic field
This question could be related to my old and Duality's newer questions.
I am building a $\mathbb{Z}/9\mathbb{Z}$ elliptic curve $E$ over $\mathbb{Q}$:
$$E: y^2+(t^3-3t^2+1)xy + t^3(t-1)^3y=x^2$$
For $...
- 913
6
votes
0
answers
219
views
Ranks of elliptic curves over cubic fields
We are writing a paper on the ranks of elliptic curves over cubic fields. The curves of different torsion subgroups are created by the formulas in Jeon et al. and by our new parametrizations.
D. Jeon,...
- 913
2
votes
1
answer
300
views
An example of a geometrically simply connected variety with infinite Brauer group (modulo constants)
$\DeclareMathOperator\Br{Br}$Let $X$ be a smooth, geometrically integral, geometrically simply connected variety over a numberfield $k$. Is it possible to have $\Br(X)/{\Br(k)}$ being an infinite ...
- 177
4
votes
2
answers
319
views
Is the set of points on an abelian surface which project to rational points on the Kummer surface a subgroup?
Let $C$ be a hyperelliptic curve of genus 2 defined over $\mathbb{Q}$, let $J$ be its Jacobian, and let $X$ be the Kummer surface associated to $J$ (i. e. $X$ is the singular Kummer surface which ...
- 658
8
votes
0
answers
231
views
Lattice point counts on the determinantal variety
I recently came across the following result of Katznelson [1]. It says that for some $C>0$, the following lattice point count holds for $n> m\geq k$.
$\#\{A \in M_{m \times n}(\mathbb{Z}) \mid \...
- 885
4
votes
2
answers
343
views
Algorithm for computing rational points if the rank of Jacobian is 0
Is there a general algorithm that can compute in finite time all rational points on any curve of genus $g\geq 2$ whose Jacobian has rank $0$?
If not, for what special cases such algorithm is known? ...
- 7,182
3
votes
0
answers
170
views
Smoothness of height in Manin conjecture
Set up: Let $K$ be a number field. Let $M_K$ be the places of $K$, and define the standard height on $\mathbb{P}^n(K)$ as
$$H([x_0, \cdots, x_n]) = \prod_{v \in M_K} \max\{|x_0|_v, \cdots, |x_n|_v\}$$
...
- 267
3
votes
1
answer
303
views
Leading constant in Batyrev-Tschinkel's refinement of Manin conjecture
Background: Let $X$ be a Fano variety over number field $K$, where its anticanonical bundle $K_X^{-1}$ is ample. Let $i: X \to \mathbb{P}^n$ be the anticanonical embedding, where $K_X^{-m} \cong i_*O(...
- 267
0
votes
0
answers
46
views
Educated guess for algebraic approximation
I found a very neat ancient hindi formula for approximating square roots using rational numbers. After doing some algebra on the formula, i came across with this recursive relation:
Given any number $...
4
votes
1
answer
917
views
Does this conic have a rational point?
Consider the conic
$$C = \{X^2+uY^2+vZ^2=0\}\subset\mathbb{P}^2_{\mathbb{Q}(u,v)}$$
over the function field $\mathbb{Q}(u,v)$.
Does $C$ have a $\mathbb{Q}(u,v)$-rational point?
- 8,998
5
votes
1
answer
217
views
Fields of definition of conjugates
Let $k$ be a field, not necessarily algebraically closed, $G$ an affine group scheme over $k$, $H$ a subgroup of $G$, and $N$ a normal subgroup of $H$, none of them assumed to be smooth. Suppose that ...
- 12.9k
2
votes
1
answer
194
views
A variant on the Fujita invariant
Let $X$ be a Fano variety over $\mathbb{C}$. Let $D$ be a divisor on $X$. Recall that the Fujita invariant of $D$ is defined to be
$$a(D) = \inf \{ t \in \mathbb{R} : K_X + tD \text{ is effective} \}.$...
- 21.4k
2
votes
0
answers
335
views
How dense is the set of rational points of a variety?
General question: Let $W$ be a proper subvariety of an irreducible affine variety $V/K$. Under what conditions do we know that $W(K)$ is a proper subset of $V(K)$?
If $K$ is finite, then one can bound ...
- 20.2k
2
votes
1
answer
242
views
Classification of quartic surfaces
Let $k$ be a field of characteristic zero (non necessarily algebraically closed, we may assume for instance that $k = \mathbb{C}(t)$). Does there exist a classification of degree four surfaces $S\...
- 8,998
1
vote
1
answer
77
views
Fixed points of rational continuous piecewise affine maps
Say that a compact convex polytope is rational if is the intersection of half-spaces whose bounding hyperplanes are the zero-sets of affine functions of the coordinates with rational coefficients. Say ...
- 19.7k
1
vote
1
answer
149
views
Geometry of contracted divisors
Let $f:\mathbb{P}^3\dashrightarrow\mathbb{P}^2$ be a dominant rational map defined over a field $k$ (not necessarily algebraically closed) of characteristic zero.
Consider a resolution $\widetilde{f}:...
- 8,998
7
votes
1
answer
296
views
Reference request. Finiteness of the Selmer group
Let $K$ be a global field (ie either a number field or the function field of a curve over a finite field). Let $A,B$ be abelian varieties over $K$ and let $\phi:A\to B$ be an isogeny. Associated with $...
- 3,076
5
votes
1
answer
286
views
How to make Burnside's formula compatible with point counting for varieties over finite fields?
If $G$ is a finite group acting on a finite set $X$, we have Burnside's formula that counts the number of orbits $|X/G|$ as:
$$ |X/G| = \frac1{|G|} \sum_{g\in G} |X^g|,
$$
with $X^g$ being the set of ...
- 167
-1
votes
1
answer
131
views
Does this quadratic system admit an integral or a rational solution?
Let $a,b$ be coprime and say $0<a<b<2a$.
Consider the quadratic system:
$$\alpha\delta-\beta\gamma=1$$
$$(\alpha^2-(\alpha\delta+\beta\gamma))a^2b+\beta^2b^3+(2\alpha\beta-\beta\delta)ab^2-\...
- 13.9k
4
votes
0
answers
129
views
Statistics about existence of rational points on a curve over $\mathbb{F}_q$
I wish to ask the naive question: if we write down a random curve $C$ over $\mathbb{F}_q$, what can be said about the probability that $C(\mathbb{F}_q)=\emptyset$?
Of course, this depends on the ...
- 1,856
11
votes
1
answer
646
views
Non-trivial solutions for $-(c^2-d^2)(a^2-b^2)=2(ad-bc)(bd+ac)$?
Consider the quartic system in four variables $a,b,c,d\in\mathbb R$:
$$-(c^2-d^2)(a^2-b^2)=2(ad-bc)(bd+ac).$$
Does this system admit rational solution with $$abcd(c^2-d^2)(a^2-b^2)(a^2-c^2)(b^2-d^2)\...
- 13.9k
7
votes
1
answer
568
views
Field extensions over which algebraic varieties cannot acquire points
The following fact (slightly reworded here) is proven in this answer:
If $K$ is a purely transcendental extension of an infinite¹ field $k$, then whenever a separated scheme $X$ of finite type over $...
- 32.5k
2
votes
1
answer
260
views
Rational points on a special class of surfaces
Consider a smooth surface of the following form
$$
S = \{f(x,y,t) = p_0(t)x^2+p_1(t)xy+p_2(t)x+p_3(t)y^2+p_4(t)y+p_5(t) = 0\}\subset\mathbb{A}^3
$$
over $\mathbb{Q}$, and set
$$
U_S = \{t' \in \mathbb{...
- 8,998
3
votes
1
answer
401
views
Rational points of bounded height on a variety
I would like to ask for some clarification on the following argument which I can not quite understand.
There is a variety $X$ of dimension $n$ over a number field with a degree two map $f:X\...
- 8,998
4
votes
1
answer
252
views
Finding the $K=\mathbb{Q}(\sqrt{6})$-rational points on the twist of $X_{0}(26)$
Let $K=\mathbb{Q}(\sqrt{6})$. I am looking to determine all $K$-rational points on the curve
$$C: y^{2}=3x^6-24x^5+24x^4-54x^3+24x^2-24x+3.$$
More precisely, $C$ is a twist of the modular curve $X_{0}(...
- 83
6
votes
1
answer
788
views
Singular curves of genus 1
Let $C$ be an irreducible curve of arithmetic genus $1$ over a field $k$ and with a double $k$-point $p\in C$.
Is $C$ rational over $k$?
If $C$ is a plane cubic the answer is positive since we can ...
- 8,998
3
votes
1
answer
719
views
Number of points of a quadric hypersurface over a finite field
Let $k = \mathbb{F}_q$ be a finite field with $q$ elements and $Q\subset\mathbb{P}^n_k$ a quadric hypersurface defined over $k$.
By the Chevalley-Warning theorem if $n\geq 2$ then $Q$ has a point. Is ...
- 8,998
3
votes
1
answer
309
views
Smooth surfaces in positive characteristic
Let $K = \mathbb{F}_p$ be a field of positive characteristic $p > 0$. Consider a surface in $\mathbb{A}^3_K$ of the following form
$$
S = \{f_1(x_0)y_0^2+f_2(x_0)y_0y_1+f_3(x_0)y_0+f_4(x_0)y_1^2+...
user114666
4
votes
1
answer
322
views
Del Pezzo surfaces of degree four and complete intersections of two quadrics
Let $X = Q_1\cap Q_2$ be a complete intersection of two smooth quadrics, over a field $K$, in $\mathbb{P}^4$ with homogeneous coordinates $y_0,y_1,y_2,y_3,y_4$.
Set $Q_1 = \{F_1 = 0\}$ and $Q_2 = \{...
- 8,998
2
votes
1
answer
326
views
$2$-isogenous to a curve in the Tate normal form
It is well-known that an elliptic curve $E$ that has a point of order $2$ and is represented as $E=[0,a,0,b,0]$ has a $2$-isogenous curve $E^\prime=[0,-2a,0,a^2-4b,0]$, see e.g. p. 507 in
A. Dujella, ...
- 913
4
votes
0
answers
195
views
Rational points on ramified coverings of abelian varieties
Let $K$ be a number field $A$ an abelian variety over $K$ and $f: X \to A$ a possibly ramified covering of degree $d$ with $X$ a proper variety. My question is:
Suppose that $f(X(K)) \neq A(K)$, can ...
- 2,224
1
vote
1
answer
175
views
Space of rational conics
Let $K$ be a field of characteristic different from two. Conics over $K$ (that is curves of degree two in $\mathbb{P}^2_K$) are parametrized by $\mathbb{P}(k[x,y,z]_2) = \mathbb{P}^5_K$.
Conisider the ...
- 8,998
7
votes
2
answers
605
views
ℤ/18ℤ elliptic curves over cubic fields
I am working on $\mathbb{Z}/18\mathbb{Z}$ elliptic curves over cubic fields. The curves are created using the formulas on p. 584 of
D. Jeon, C. H. Kim, Y. Lee, Families of elliptic curves over cubic ...
- 913
1
vote
0
answers
298
views
Surjectivity of a norm map over $ \mathbb{Q} $
Suppose $ (L/L') $ is an galois extension , where both fields are extension of $Q$ of $\dim n $ and $\dim n^{'}$ respectively.Suppose we consider the norm map $ Nr_{(L/L')} :L \rightarrow L' $. ...
- 923
0
votes
1
answer
199
views
Maximum dimension of a simultaneous anisotropic subspace of quadratic forms over $ \mathbb{Q} $
Let $(V,q )$ be a quadratic space over $ \mathbb{Q} $. A subspace $ U $ is called totally isotropic if $ q(x) = 0 $ for all $ x \in U $ and a subspace $ U $ is called an anisotropic subspace if $ ...
- 923
3
votes
1
answer
173
views
Linear subspace in quadric hypersurfaces over a field
Let $K$ be a field of characteristic different from two, and $Q\subset\mathbb{P}^{n+1}_K$ an $n$-dimensional smooth quadric hypersurface over $K$.
Suppose also that $Q$ has a $K$-point and so $Q$ is ...
user58604