Questions tagged [rational-points]

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16 votes
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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 ...
Asvin's user avatar
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12 votes
0 answers
656 views

Kihara-like Z/6Z elliptic curve families

Shoichi Kihara constructed a family of elliptic curves with Mordell–Weil group $\mathbb{Z}/6\mathbb{Z}\times\mathbb{Z}^3$ (generic rank at least 3) in 2006. Kihara's family produces a number of rank 8 ...
Maksym Voznyy's user avatar
12 votes
0 answers
350 views

Artin representations appearing in Mordell-Weil groups of elliptic curves

Let $E$ be an elliptic curve defined over $\mathbf{Q}$, and let $K$ be a Galois number field. The Galois group $G=\mathrm{Gal}(K/\mathbf{Q})$ acts on the Mordell-Weil group $E(K)$ and thus on the ...
François Brunault's user avatar
10 votes
0 answers
211 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
  • 1,989
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
8 votes
0 answers
129 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{...
user avatar
7 votes
0 answers
203 views

Lattice radial-step (ratchet) spirals

(30Oct13: Now solved; see Addendum.) Define a curve, a ratchet spiral, $S(r_0,\epsilon)$ as follows, where $r_0 > 0$ and $\epsilon < 1$.     $S(r_0,\epsilon)$ begins with the arc ...
Joseph O'Rourke'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
6 votes
0 answers
209 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
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6 votes
0 answers
113 views

Rational $d$-simplices

Define a rational $d$-simplex as a simplex in $\mathbb{R}^d$ such that the measure of all its $k$-dimensional faces, $k \ge 1$, is rational. So a rational triangle has rational edge lengths and ...
Joseph O'Rourke's user avatar
6 votes
0 answers
435 views

Brauer-Manin obstruction to surfaces of Kodaira dimension 1

Roughly speaking, the Kodaira dimension is an invariant of a variety that corresponds to curvature. One can show that curves of genus $\geq 2$ have Kodaira dimension 1 using Riemann-Roch. In Corollary ...
Jackson Morrow's user avatar
5 votes
0 answers
290 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
5 votes
0 answers
184 views

Given a point $P$ on a genus-$1$ curve over $\mathbb{Q}_p$, is there an $R$ such that $2 \mid [P - R]$ and $x(\overline{P}) \neq x(\overline{R})$?

Let $p \in \mathbb{Z}$ be prime, and let $f \in \mathbb{Z}_p[x]$ be a quartic polynomial with nonzero discriminant. Let $C/\mathbb{Q}_p$ be the genus-$1$ curve with affine equation $y^2 = f(x)$. Let $\...
IMT's user avatar
  • 53
5 votes
0 answers
149 views

Counting square zero forms over finite fields

Let $p$ be an odd prime and let $R=\Lambda_{\mathbb{F}_p}[x_1,\dots,x_n]$ be the exterior algebra on $n$ generators over the finite field with $p$ elements. This is a graded-commutative ring. Is ...
KotelKanim's user avatar
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5 votes
0 answers
292 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, ...
Yonatan Harpaz's user avatar
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
4 votes
0 answers
173 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
4 votes
0 answers
235 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
4 votes
0 answers
298 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 ...
Sam Streeter's user avatar
4 votes
0 answers
1k views

Rational points on the unit circle

Is anything known about any of the following questions about rational points on the unit circle? By “double point” I mean an element of $2C$, where $C$ is the group of rational points on the unit ...
Robin Houston'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
0 answers
311 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 $\mathbb{Z}$-...
user avatar
3 votes
0 answers
542 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 ...
Stanley Yao Xiao's user avatar
3 votes
0 answers
269 views

The uniform boundedness of rational torsion for traceless abelian surfaces over a function field

The function field analog of the theorem of Mazur-Kamienny-Merel (giving a universal bound in $[K:\mathbb{Q}]$ for the size of the $K$-rational torsion of an elliptic curve) is immediate just from the ...
Vesselin Dimitrov's user avatar
3 votes
0 answers
308 views

Rational points and Tesla cards

I'm rapidly approaching 300,000 curves in my ongoing search for Mordell curves of rank >=8. Currently I'm finding that I have a bottleneck in the code that locates rational points on these curves. ...
Kevin Acres's user avatar
2 votes
0 answers
110 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 $...
Maksym Voznyy's user avatar
2 votes
0 answers
190 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
  • 19.3k
2 votes
0 answers
263 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
2 votes
0 answers
190 views

Finding rational points via birational map

Let $C$ be an affine curve given by $p_C(x,y)=0$ where $$ p_C=2x^3y + 2xy^3 +x^3 + y^3 + 5x^2y + 5xy^2 + 2x^2 + 2y^2 + 2x^2y^2 + 2xy $$ and let $\overline{C}$ denote the projective closure of $C$. For ...
monoid911's user avatar
2 votes
0 answers
304 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 ...
Max CYLin's user avatar
  • 151
2 votes
0 answers
351 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 $...
Hidegol's user avatar
  • 21
2 votes
0 answers
105 views

the least point on a variety over a finite field

Let $p$ be a large prime parameter and $V\subseteq \mathbb{P}^n_{\mathbb{F}_p}$ a variety defined over the finite field $\mathbb{F}_p$ with bounded degree and dimension (w.r.t. $p$). Assume that $V$ ...
Lior Bary-Soroker's user avatar
2 votes
0 answers
120 views

Benchmark problems for computing rational points on varieties

Are there standard benchmark problem sets used for empirically evaluating algorithms designed for computing rational points on (various classes of) algebraic varieties? If so, could you please point ...
Grant Olney Passmore's user avatar
1 vote
0 answers
241 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
1 vote
0 answers
183 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
1 vote
0 answers
138 views

Properties of pointless projective curves over finite fields?

Probably not research level, feel free to downvote. We got construction of bounded degree projective curves with no points over finite fields. This construction generalizes to higher dimension. One of ...
joro's user avatar
  • 24.2k
1 vote
0 answers
122 views

Rational points of torsors over a separable closure

I already asked this question on Math Stack few days ago ( torsors over a separable closure ), but did not receive any answer, so I post it here. Let $G$ be a smooth linear algebraic group defined ...
GreginGre's user avatar
  • 1,661
1 vote
0 answers
169 views

Levi decompositions of k-rational points of linear algebraic groups

Let $k$ be a field with characteristic zero and $G$ be a (connected or not) linear algebraic group defined over $k$. We know that $G$ has a Levi decomposition i.e., $G=R_u(G)\rtimes L$, where $R_u(G)$ ...
m07kl's user avatar
  • 1,672
1 vote
0 answers
119 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 $K$-...
Jimmy's user avatar
  • 545
1 vote
0 answers
96 views

Shortest paths stepping on rational points of height $h$

Q. Do shortest paths walking between rational points of height $h$ ever properly cross themselves? Explaining this question takes a bit of definitional exposition. First, I copy definitions from ...
Joseph O'Rourke's user avatar
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 ...
Keesjan's user avatar
  • 51
0 votes
0 answers
41 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
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
0 answers
249 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 ...
user36371's user avatar
  • 101
0 votes
0 answers
208 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,\...
Jana's user avatar
  • 2,022
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