Questions tagged [rational-points]

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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 $...
Maksym Voznyy's user avatar
6 votes
0 answers
195 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,...
Maksym Voznyy's user avatar
2 votes
1 answer
276 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 ...
Victor de Vries's user avatar
4 votes
2 answers
302 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 ...
Vik78's user avatar
  • 487
8 votes
0 answers
224 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 \...
Breakfastisready's user avatar
4 votes
2 answers
302 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? ...
Bogdan Grechuk's user avatar
3 votes
0 answers
157 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 $$H([x_0, \cdots, x_n]) = \prod_{v \in M_K} \max\{|x_0|_v, \cdots, |x_n|_v\}$$ ...
dummy's user avatar
  • 257
3 votes
1 answer
260 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(...
dummy's user avatar
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0 votes
0 answers
42 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 $...
Simón Flavio Ibañez's user avatar
4 votes
1 answer
853 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?
Puzzled's user avatar
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5 votes
1 answer
192 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 ...
LSpice's user avatar
  • 11.4k
2 votes
1 answer
164 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} \}.$...
Daniel Loughran's user avatar
2 votes
0 answers
272 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 ...
H A Helfgott's user avatar
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2 votes
1 answer
225 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\...
Puzzled's user avatar
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1 vote
1 answer
72 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 ...
James Propp's user avatar
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1 vote
1 answer
148 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}:...
Puzzled's user avatar
  • 8,852
7 votes
1 answer
274 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 $...
Damian Rössler's user avatar
5 votes
1 answer
282 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 ...
bernardorim's user avatar
-1 votes
1 answer
129 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-\...
Turbo's user avatar
  • 13.7k
4 votes
0 answers
125 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 ...
Cheng-Chiang Tsai's user avatar
11 votes
1 answer
641 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)\...
Turbo's user avatar
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7 votes
1 answer
499 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 $...
Gro-Tsen's user avatar
  • 30.2k
2 votes
1 answer
254 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{...
Puzzled's user avatar
  • 8,852
3 votes
1 answer
352 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\...
Puzzled's user avatar
  • 8,852
4 votes
1 answer
238 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}(...
Maleeha's user avatar
  • 43
6 votes
1 answer
698 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 ...
Puzzled's user avatar
  • 8,852
3 votes
1 answer
565 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 ...
Puzzled's user avatar
  • 8,852
3 votes
1 answer
297 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+...
user avatar
4 votes
1 answer
292 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 = \{...
Puzzled's user avatar
  • 8,852
2 votes
1 answer
309 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, ...
Maksym Voznyy's user avatar
4 votes
0 answers
175 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 ...
Maarten Derickx's user avatar
1 vote
1 answer
164 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 ...
Puzzled's user avatar
  • 8,852
7 votes
2 answers
569 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 ...
Maksym Voznyy's user avatar
1 vote
0 answers
247 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' $. ...
Sky's user avatar
  • 913
0 votes
1 answer
181 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 $ ...
Sky's user avatar
  • 913
2 votes
1 answer
150 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 ...
user avatar
3 votes
1 answer
252 views

Rationalizing and minimizing elliptic curve coefficients

I am working on elliptic curves with torsion group $\mathbb{Z}/14\mathbb{Z}$ over quadratic fields. The curves are constructed using the model $E_1=[0,a,0,b,0]$ following the formulas on p. 13 of L. ...
Maksym Voznyy's user avatar
0 votes
0 answers
86 views

Computational tool for checking the existence of non-trivial rational zero of a cubic form

Suppose we consider a arbitrary cubic homogeneous form $f$ in in four or five variables over the rational field. Is there any computational tool or algorithm to check whether this cubic homogeneous ...
Sky's user avatar
  • 913
0 votes
1 answer
350 views

Systems of equations for elliptic curves without $3$-torsion

In his YouTube video New rank records for elliptic curves having rational torsion, Noam Elkies uses systems of equations at 6:16 and 8:38 to present $\mathbb{Z}/3\mathbb{Z}$ curves of rank 14 and rank ...
Maksym Voznyy's user avatar
5 votes
1 answer
505 views

Lines on quadric surfaces

Consider a smooth quadric surface $Q\subset\mathbb{P}^3$ over a field $k$. Are there natural hypotheses one can put on $k$ in order to ensure the existence of a line defined over $k$ on $Q$?
Puzzled's user avatar
  • 8,852
5 votes
2 answers
552 views

Birational geometry over finite fields

I apologize in advance since probably my questions are very naive. I would like to understand some central notions in birational geometry, that are clear to me over the complex numbers, for varieties ...
user avatar
7 votes
1 answer
307 views

Geometrically rational variety over a finite field

Let $k=\mathbb{F}_q$ be a finite field, and let $X$ be a smooth projective variety over $k$. Suppose that $X_{\overline{k}}$ is birational to $\mathbb{P}^n_{\overline{k}}$, do we know (1)If $X$ is ...
user avatar
2 votes
0 answers
265 views

Rational points on surfaces

Let $k$ be a field of characteristic zero. In the affine space $\mathbb{A}_{x,y,t}^3$ consider a surface $S$ of the form $$ S = \{a_0(t)x^2+a_1(t)xy+a_2(t)x+a_3(t)y^2+a_4(t)y+a_5(t) = 0\} $$ where $...
user avatar
4 votes
0 answers
240 views

Is equation $y^3+x y + x^4 + 4 = 0$ solvable locally (in ${\mathbb Q}_p$ for all $p$)?

When finding out whether an equation in 2 variables has rational solutions (or, equivalently, whether an algebraic curve has any rational points), many authors recommend checking the local solubility ...
Bogdan Grechuk's user avatar
1 vote
0 answers
184 views

Brauer-Manin obstruction and affine curves

I'm looking for references that can justify to what extent is the following statement true: Statement. Let $X$ be a smooth geometrically integral curve over a number field $k$. Then the Brauer-Manin ...
oleout's user avatar
  • 865
6 votes
0 answers
210 views

Rational points on varieties whose anticanonical bundle is nef but not ample

Is the following plausible? "If $X$ is a variety over $\mathbf{Q}$ whose anticanonical bundle $L$ is nef but not ample, there is a number field $K$ such that $X(K)$ contains an infinite set of ...
JSE's user avatar
  • 19.1k
0 votes
1 answer
130 views

Primes of the form $p=u^2+1$ and number of points on the elliptic curve $x^3+a x z^2=y^2 z$

Let $p$ be prime of the form $p=u^2+1$. For $a \in \mathbb{F}_p,a \ne 0$, define $E_a : x^3+a x z^2=y^2 z$ Let $B= \lfloor 2 \sqrt{p}\rfloor$ Must we have $(\#E_a(\mathbb{F}_p) -p - 1) \in \{2,-2,B,-B\...
joro's user avatar
  • 24.2k
-1 votes
1 answer
250 views

Bounds for the number of points on projective hyperelliptic curves over finite fields

Let $C$ be projective hyperelliptic curve over finite field $K$. What are bounds for the number of points $\#C(K)$? The Hasse-Weil bound requires smooth curves, and hyperelliptic curves are not smooth ...
joro's user avatar
  • 24.2k
5 votes
0 answers
292 views

2-descent on elliptic curves, and units modulo squares of units

Setup: Let $p$ be a prime, let $f(x) \in \mathbb{Q}_p[x]$ be a separable monic cubic polynomial cutting out the maximal order $\mathcal{O}_{K_f}$ in the etale algebra $K_f := \mathbb{Q}_p[x]/(f(x))$, ...
Ashvin Swaminathan's user avatar
10 votes
0 answers
212 views

Are the nonnegative rationals diophantine with only two quantifiers?

Definition: A subset $D\subseteq \mathbb{Q}$ is diophantine if it is the projection of the zero set of a polynomial, i.e. there exists a polynomial $f\in\mathbb{Q}[X,Y_1,\dots,Y_n]$ for some $n$ such ...
Arno Fehm's user avatar
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