Rational point
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Arithmetic degrees and dynamical degrees of endomorphisms on surfaces Open
For a dominant rational self-map on a smooth projective variety defined over\na number field, Shu Kawaguchi and Joseph H. Silverman conjectured that the\ndynamical degree is equal to the arithmetic degree at a rational point whose\nforward…
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Heights on stacks and a generalized Batyrev–Manin–Malle conjecture Open
We define a notion of height for rational points with respect to a vector bundle on a proper algebraic stack with finite diagonal over a global field, which generalizes the usual notion for rational points on projective varieties. We expla…
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Torsion points on elliptic curves over number fields of small degree. Open
Let d be an integer and let K be a number field of degree d over Q. By the Mordell- Weil theorem we know that if E is an elliptic curve over K then there exist unique integers m, n > 0 and r \geq 0 such that E(K)_{tors} = Z/mZ x Z/mnZ x Zr…
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On the distribution of rational points on ramified covers of abelian varieties Open
We prove new results on the distribution of rational points on ramified covers of abelian varieties over finitely generated fields $k$ of characteristic zero. For example, given a ramified cover $\pi : X \to A$ , where $A$ is an abelian va…
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Beyond heights: slopes and distribution of rational points Open
The distribution of rational points of bounded height on algebraic varieties is far from uniform. Indeed the points tend to accumulate on thin subsets which are images of non-trivial finite morphisms. The problem is to find a way to charac…
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Dynamical degree and arithmetic degree of endomorphisms on product varieties Open
For a dominant rational self-map on a smooth projective variety defined over a number field, Shu Kawaguchi and Joseph H. Silverman conjectured that the dynamical degree is equal to the arithmetic degree at a rational point whose forward or…
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MANY CUBIC SURFACES CONTAIN RATIONAL POINTS Open
Building on recent work of Bhargava-Elkies-Schnidman and Kriz-Li, we produce infinitely many smooth cubic surfaces defined over the field of rational numbers that contain rational points.
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<i>q</i>-deformed rational numbers and the 2-Calabi–Yau category of type Open
We describe a family of compactifications of the space of Bridgeland stability conditions of a triangulated category, following earlier work by Bapat, Deopurkar and Licata. We particularly consider the case of the 2-Calabi–Yau category of …
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On (ψ, ϕ)-Rational Contractions Open
In this paper, we examine the notion of ( ψ , ϕ )-contractions by involving rational forms in the context of complete metric spaces. We note that some well-known fixed point theorems for rational forms can be deduced from our main results.…
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ON MAHLER’S CLASSIFICATION OF -ADIC NUMBERS Open
We give transcendence measures for $p$ -adic numbers $\unicode[STIX]{x1D709}$ , having good rational (respectively, integer) approximations, that force them to be either $p$ -adic $S$ -numbers or $p$ -adic $T$ -numbers.
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The Hilbert schemes of points on surfaces with rational double point singularities Open
We prove that the Hilbert scheme of points on a normal quasi-projective surface with at worst rational double point singularities is irreducible.
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Level curves of rational functions nd unimodular points on rational curves Open
We obtain an improvement and broad generalisation of a result of N. Ailon and Z. Rudnick (2004) on common zeros of shifted powers of polynomials. Our approach is based on reducing this question to a more general question of counting inters…
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Khovanskii-finite valuations, rational curves, and torus actions Open
We study full rank homogeneous valuations on (multi)-graded domains and ask when they have finite Khovanskii bases. We show that there is a natural reduction from multigraded to simply graded domains. As special cases, we consider projecti…
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Uniform bounds for the number of rational points on hyperelliptic curves of small Mordell–Weil rank Open
We show that there is a bound depending only on g, r and [ K : \mathbb Q ] for the number of K -rational points on a hyperelliptic curve C of genus g over a number field K such that the Mordell–Weil rank r of its Jacobian is at most g–3 . …
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Arithmetic degrees and dynamical degrees of endomorphisms on surfaces Open
For a dominant rational self-map on a smooth projective variety defined over a number field, Kawaguchi and Silverman conjectured that the (first) dynamical degree is equal to the arithmetic degree at a rational point whose forward orbit is…
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The proportion of genus one curves over ℚ defined by a binary quartic that everywhere locally have a point Open
We consider the proportion of genus one curves over [Formula: see text] of the form [Formula: see text] where [Formula: see text] is a binary quartic form (or more generally of the form [Formula: see text] where also [Formula: see text] is…
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An elliptic sequence is not a sampled linear recurrence sequence Open
Let $E$ be an elliptic curve defined over the rationals and in minimal Weierstrass form, and let $P=(x_1/z_1^2,y_1/z_1^3)$ be a rational point of infinite order on $E$, where $x_1,y_1,z_1$ are coprime integers. We show that the integer seq…
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Cyclotomic and abelian points in backward orbits of rational functions Open
We prove several results on backward orbits of rational functions over number fields. First, we show that if K is a number field, ϕ∈K(x) and α∈K then the extension of K generated by the abelian points (i.e. points that generate an abelian …
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On the characterization of rational homotopy types and Chern classes of closed almost complex manifolds Open
We give an exposition of Sullivan’s theorem on realizing rational homotopy types by closed smooth manifolds, including a discussion of the necessary rational homotopy and surgery theory, adapted to the realization problem for almost comple…
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The perfect power problem for elliptic curves over function fields Open
We generalise the Siegel-Voloch theorem about S-integral points on elliptic curves as follows: let K/ F denote a global function field over a finite field F of characteristic p ≥ 5, let S denote a finite set of places of K and let E/K deno…
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$q$-deformed rational numbers and the 2-Calabi--Yau category of type $A_2$ Open
We describe a family of compactifications of the space of Bridgeland stability conditions of any triangulated category following earlier work by Bapat, Deopurkar, and Licata. We particularly consider the case of the 2-Calabi--Yau category …
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Heights on stacks and a generalized Batyrev-Manin-Malle conjecture Open
We define a notion of height for rational points with respect to a vector bundle on a proper algebraic stack with finite diagonal over a global field, which generalizes the usual notion for rational points on projective varieties. We expla…
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Odd degree isolated points on $X_1(N)$ with rational $j$-invariant Open
Let $C$ be a curve defined over a number field $k$. We say a closed point $x\in C$ of degree $d$ is isolated if it does not belong to an infinite family of degree $d$ points parametrized by the projective line or a positive rank abelian su…
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Reconstruction of rational polytopes from the real-parameter Ehrhart function of its translates Open
When extending the Ehrhart lattice point enumerator $L_P(t)$ to allow real dilation parameters $t$, we lose the invariance under integer translations that exists when $t$ is restricted to be an integer. This paper studies this phenomenon; …
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Diophantine Approximations on Definable Sets Open
Consider the vanishing locus of a real analytic function on $\mathbb{R}^n$ restricted to $[0,1]^n$. We bound the number of rational points of bounded height that approximate this set very well. Our result is formulated and proved in the co…
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Iterated towers of number fields by a quadratic map defined over the Gaussian rationals Open
An iterated tower of number fields is constructed by adding preimages of a base point by iterations of a rational map. A certain basic quadratic rational map defined over the Gaussian number field yields such a tower of which any two steps…
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The proportion of genus one curves over $\mathbb{Q}$ defined by a binary quartic that everywhere locally have a point Open
We consider the proportion of genus one curves over $\mathbb{Q}$ of the form $z^2=f(x,y)$ where $f(x,y)\in\mathbb{Z}[x,y]$ is a binary quartic form (or more generally of the form $z^2+h(x,y)z=f(x,y)$ where also $h(x,y)\in\mathbb{Z}[x,y]$ i…
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Rational Points on Elliptic K3 Surfaces of Quadratic Twist Type Open
In studying rational points on elliptic K3 surfaces of the form $$\begin{equation*} f(t)y^2=g(x), \end{equation*}$$ where f, g are cubic or quartic polynomials (without repeated roots), we introduce a condition on the quadratic twists of t…
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Rationally connected varieties over the maximally unramified extension of $p$-adic fields Open
A result of Graber, Harris and Starr shows that a rationally connected variety defined over the function field of a curve over the complex numbers always has a rational point. Similarly, a separably rationally connected variety over a fini…
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The number of varieties in a family which contain a rational point Open
We consider the problem of counting the number of varieties in a family over a number field which contain a rational point. In particular, for products of Brauer–Severi varieties and closely related counting functions associated to Brauer …