# Questions tagged [rational-points]

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### $2$-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, ...
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### 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 ...
1 vote
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### 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 ...
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### 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 ...
1 vote
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### 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 $Nr_{(L/L')} :L \rightarrow L'$. ...
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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 $... 2 votes 1 answer 78 views ### Linear subspace in quadric hypersurfaces over a field Let$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$is ... 3 votes 1 answer 222 views ### Rationalizing and minimizing elliptic curve coefficients I am working on elliptic curves with torsion group$\mathbb{Z}/14\mathbb{Z}$over quadratic fields. The curves are constructed using the model$E_1=[0,a,0,b,0]$following the formulas on p. 13 of L. ... 0 votes 0 answers 82 views ### Computational tool for checking the existence of non-trivial rational zero of a cubic form Suppose we consider a arbitrary cubic homogeneous form$f$in in four or five variables over the rational field. Is there any computational tool or algorithm to check whether this cubic homogeneous ... 0 votes 1 answer 306 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 ... 5 votes 1 answer 186 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$Q$? 5 votes 2 answers 494 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 ... 6 votes 1 answer 203 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 ... 2 votes 0 answers 227 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$... 176 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 ...
1 vote
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### 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 ...
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### 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 ...
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### Why should the number of $\mathbb{F}_q$ points on degree $d$ curves $C\subset \mathbb{P}_{\mathbb{F}_q}^n$ decrease as $n$ increases?

This question concerns some counterintuitive results (to me at least) regarding the number of points on a projective curve over a finite field. Namely, if one fixes the degree of the curve, but ...
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 $C\to\mathbb{... 3 votes 1 answer 317 views ### Finding$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)$is a hyperelliptic curve of genus 3 and has the simplified affine model: \... 3 votes 2 answers 170 views ### primes of multiplicative reduction for elliptic curves with rational$\ell$-torsion Very recently, I made an observation from scanning lists of elliptic curves on the LMFDB that leads me to the following (unproven) statement: Fix$\ell \in \{5, 7\}$. Let$E$be an elliptic curve ... 0 votes 1 answer 451 views ### How I can prove or disprove that$\frac{x}{y+z}+\frac{y}{x+z}+\frac{z}{y+x}=1\$ has solutions in rationals? [duplicate] 