Let $q$ be a perfect power of a prime number $p$ and $E({\mathbb F}_q)$ be an elliptic curve over ${\mathbb F}_q$ given by the equation $y^2=x^3+Ax+B$. For a positive integer $n$ we denote by $ \# E({\mathbb F}_{q^n})$ the number of rational points on $E$ (including infinity) over the extension ${\mathbb F}_{q^n}$. Under a mild technical condition, we show that the sequence $\lbrace \# E({\mathbb F}_{q^n}) \rbrace_{n>0}$ contains at most $10^{200}$ perfect squares. If the mild condition is not satisfied, then $\#E({\mathbb F}_{q^n})$ is a perfect square for infinitely many $n$ including all the multiples of $24$. Our proof uses a quantitative version of the Subspace Theorem. We also find all the perfect squares for all such sequences in the range $q < 50$ and $n\leq 1000$.
We show by finding an explicit parametrization that a 4th degree surface which arises as a necessary condition for the existence of a perfect cuboid is a rational surface, i.e. birationally equivalent over $\mathbb Q$ to a plane.
A perfect cuboid is a rectangular parallelepiped with integer edges and integer face diagonals whose space diagonal is also integer. The existence of such cuboids is neither proved, nor disproved. A rational perfect cuboid is a natural companion of a perfect cuboid absolutely equivalent to the latter one. Its edges and face diagonals are rational numbers, while its space diagonal is equal to unity. Recently, based on a symmetry reduction, it was shown that edges of a rational perfect cuboid are roots of a certain cubic equation with rational coefficients depending on two rational parameters. F…
A rational perfect cuboid is a rectangular parallelepiped whose edges and face diagonals are given by rational numbers and whose space diagonal is equal to unity. Finding such a cuboid is equivalent to finding a perfect cuboid with all integer edges and diagonals, which is an old unsolved problem. Recently, based on a symmetry approach, it was shown that edges and face diagonals of rational perfect cuboid are roots of two cubic equations whose coefficients depend on two rational parameters. Six special cases where these cubic equations are reducible have been already found. Two more possible c…
It is well known that the arithmetic nature of Mills' prime-representing constant is uncertain: we do not know if Mills' constant is a rational or irrational number. In the case of other prime-representing constants, irrationality can be proved, but it is not known whether these constants are algebraic or transcendental numbers. By using Liouville or Roth's theorems about approximation by rationals, we find a couple of prime-representing constants that can be proved to be transcendental numbers.
Two cubic equations and three auxiliary equations for edges and face diagonals of a rational perfect cuboid have been recently derived. They constitute a background for two inverse problems. The coefficients of cubic equations and the right hand sides of auxiliary equations are rational functions of two rational parameters, i. e. they have denominators. Hence the inverse problems have singular points. These singular points are studied in the present paper.
A rational perfect cuboid is a rectangular parallelepiped whose edges and face diagonals are given by rational numbers and whose space diagonal is equal to unity. Its existence is equivalent to the existence of a perfect cuboid with all integer edges and diagonals. Finding such a cuboid or proving its non-existence is an old unsolved problem. Recently, based on a symmetry approach, the equations of a perfect cuboid were transformed to factor equations. The factor equations turned out to be solvable and, being solved, have led to a pair of inverse problems. Our efforts in the present paper are …
Let $\mathbb K$ be a perfect field of characterstic $p\ge 0$ and let $R\in \mathbb K(x)$ be a rational function. This paper studies the number $Δ_{α, R}(n)$ of distinct solutions of $R^{(n)}(x)=α$ over the algebraic closure $\overline{\mathbb K}$ of $\mathbb K$, where $α\in \overline{\mathbb K}$ and $R^{(n)}$ is the $n$-fold composition of $R$ with itself. With the exception of some pairs $(α, R)$, we prove that $Δ_{α, R}(n)=c_{α, R}\cdot d^n+O_{α, R}(1)$ for some $0<c_{α, R}\le 1<d$. The number $d$ is readily obtained from $R$ and we provide estimates on $c_{α, R}$. Moreover we prove that the…
On the twisted Fermat cubic, an elliptic divisibility sequence arises as the sequence of denominators of the multiples of a single rational point. It is shown that there are finitely many perfect powers in such a sequence whose first term is greater than 1. Moreover, if the first term is divisible by 6 and the generating point is triple another rational point then there are no perfect powers in the sequence except possibly an lth power for some l dividing the order of 2 in the first term.
A rational Diophantine m-tuple is a set of m nonzero rationals such that the product of any two of them increased by 1 is a perfect square. The first rational Diophantine quadruple was found by Diophantus, while Euler proved that there are infinitely many rational Diophantine quintuples. In 1999, Gibbs found the first example of a rational Diophantine sextuple, and in 2016 Dujella, Kazalicki, Mikić and Szikszai proved that there are infinitely many of them. In this paper, we prove that there exist infinitely many rational Diophantine sextuples such that the denominators of all the elements in …
A rational perfect cuboid is a rectangular parallelepiped whose edges and face diagonals are given by rational numbers and whose space diagonal is equal to unity. Recently it was shown that the Diophantine equations describing such a cuboid lead to a couple of parametric families of elliptic curves. Two and three descent methods for calculating their ranks are discussed in the present paper. The elliptic curves in each parametric family are subdivided into two subsets admitting 2-descent and 3-descent methods respectively.
A perfect cuboid (PC) is a rectangular parallelepiped with rational sides $a,b,c$ whose face diagonals $d_{ab}$, $d_{bc}$, $d_{ac}$ and space (body) diagonal $d_s$ are rationals. The existence or otherwise of PC is a problem known since at least the time of Leonhard Euler. This research establishes equivalent conditions of PC by nontrivial rational solutions $(X,Y)$} and $(Z,W)$} of congruent number equation $ y^2=x^3-N^2x$, where product $XZ$ is a square. By using such pair of solutions five parametrizations of nearly-perfect cuboid (NPC) (only one face diagonal is irrational) and five equiva…
Research Center in Mathematics and Applications (CIMA)（数学与应用研究中心）；Department of Mathematics, School of Sciences and Technology of the University of Évora（埃武拉大学科学与技术学院数学系）
本文研究了素数表示常数递归构造中出现的映射$\mathcal{T}(x)=\lfloor x floor(1+\{x\})$的有理动力学，定义了有理数的阶，并证明了有限阶有理数的自然密度为1，同时给出了特定分母下的显式分数家族。
A perfect triangle is a triangle with rational sides, medians, and area. In this article, we use a similar strategy due to Pocklington to show that if $Δ$ is a perfect triangle, then it cannot be an isosceles triangle. It gives a partial answer to a question of Richard Guy, who asked whether any perfect triangles exist. No example has been found to date. It is widely believed that such a triangle does not exist.
We study real quadratic fields $\mathbb{Q}(\sqrt{D})$ such that, for a given rational integer $m$, all $m$-multiples of totally positive integers are sums of squares. We prove quite sharp necessary and sufficient conditions for this to happen. Further, we give a fast algorithm that solves this question for specific $m$, $D$ and we give complete results for $m \leq 5000$.
In this paper we define a rigid rational homotopy type, associated to any variety $X$ over a perfect field $k$ of positive characteristic. We prove comparison theorems with previous definitions in the smooth and proper, and log-smooth and proper case. Using these, we can show that if $k$ is a finite field, then the Frobenius structure on the higher rational homotopy groups is mixed. We also define a relative rigid rational homotopy type, and use it to define a homotopy obstruction for the existence of sections.
In this paper, we consider elliptic curves induced by rational Diophantine quadruples, i.e. sets of four nonzero rationals such that the product of any two of them plus 1 is a perfect square. We show that for each of the groups $\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/k\mathbb{Z}$ for $k = 2, 4, 6, 8$, there are infinitely many rational Diophantine quadruples with the property that the induced elliptic curve has this torsion group. We also construct curves with moderately large rank in each of these four cases.
Rational Diophantine triples, i.e. rationals a,b,c with the property that ab+1, ac+1, bc+1 are perfect squares, are often used in construction of elliptic curves with high rank. In this paper, we consider the opposite problem and ask how small can be the rank of elliptic curves induced by rational Diophantine triples. It is easy to find rational Diophantine triples with elements with mixed signs which induce elliptic curves with rank 0. However, the problem of finding such examples of rational Diophantine triples with positive elements is much more challenging, and we will provide the first su…
In this note, we prove the existence of a secondary term in the asymptotic formula of the cubic moment of quadratic Dirichlet L-functions $$\sum_{\substack{d - \mathrm{monic \, \& \, sq. \, free} \mathrm{deg}\, d \, = \, D}} L\big(\tfrac{1}{2}, χ_{d}\big)^{3}$$ over rational function fields on the order of $q^{\scriptscriptstyle \frac{3}{4} D}.$ This term is in perfect analogy with the $x^{\scriptscriptstyle \frac{3}{4}}$-term indicated in our joint work arXiv:math/0110092v1 for the corresponding asymptotic formula over the rationals.
A rational perfect cuboid is a rectangular parallelepiped whose edges and face diagonals are given by rational numbers and whose space diagonal is equal to unity. It is described by a system of four equations with respect to six variables. The cuboid factor equations were derived from these four equations by symmetrization procedure. They constitute a system of eight polynomial equations, which has been solved parametrically. In the present paper its parametric solution is expressed through intersections or rational and elliptic curves arranged into parametric families.
A rational positive-definite quadratic form is perfect if it can be reconstructed from the knowledge of its minimal nonzero value m and the finite set of integral vectors v such that f(v) = m. This concept was introduced by Voronoi and later generalized by Koecher to arbitrary number fields. One knows that up to a natural "change of variables'' equivalence, there are only finitely many perfect forms, and given an initial perfect form one knows how to explicitly compute all perfect forms up to equivalence. In this paper we investigate perfect forms over totally real number fields. Our main resu…
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>3, let S denote a finite set of places of K and let E/K denote a non-isotrivial elliptic curve over K with j-invariant a p^s power but not p^(s+1) power in K. Fix a non-constant function f in K(E) with a pole of order N>0 at the zero element of E. We prove that there are only finitely many rational points P in E(K) such that for any valuation outside S for which f(P) is negative, that valuation of f(P) is divisible by so…
Let S be a sphere in R^n such that S\capQ^n\neq\emptyset and let Cl denote the closure operator in the Euclidean topology of R^n. If the center of S is in Q^n, then Cl(S\capQ^n) is S, as is easily proved. If the center of S is not in Q^n, then what is Cl(S\capQ^n)? This question, which was answered partially in the author's paper [Proc. Japan Acad. Ser. A Math. Sci. 80 (2004), no. 7, 146--149], is answered completely in this paper by representing Cl(S\capQ^n) in terms of the group of Q-automorphisms of the algebraic closure of Q(γ_1,...,γ_n) in C, where γ_1,...,γ_n denote the coordinates of th…
A rational Diophantine triple is a set of three nonzero rational a,b,c with the property that ab+1, ac+1, bc+1 are perfect squares. We say that the elliptic curve y^2 = (ax+1)(bx+1)(cx+1) is induced by the triple {a,b,c}. In this paper, we describe a new method for construction of elliptic curves over Q with reasonably high rank based on a parametrization of rational Diophantine triples. In particular, we construct an elliptic curve induced by a rational Diophantine triple with rank equal to 12, and an infinite family of such curves with rank >= 7, which are both the current records for that k…
A conjecture of Serre concerns the number of rational points of bounded height on a finite cover of projective space P^{n-1}. In this paper, we achieve Serre's conjecture in the special case of smooth cyclic covers of any degree when n is at least 10, and surpass it for covers of degree 3 or higher when n > 10. This is achieved by a new bound for the number of perfect r-th power values of a polynomial with nonsingular leading form, obtained via a combination of an r-th power sieve and the q-analogue of van der Corput's method.
We prove the $W\mathcal{O}$-rationality of klt threefolds and the rational chain connectedness of klt Fano threefolds over a perfect field of characteristic $p>5$. As a consequence, any klt Fano threefold over a finite field has a rational point.
We construct a family of ideals representing ideal classes of order 2 in quadratic number fields and show that relations between their ideal classes are governed by certain cyclic quartic extensions of the rationals.
We study representation of square-free polynomials in the polynomial ring F[t] over a finite field F by polynomials in F[t][x]. This is a function field version of the well-studied problem of representing squarefree integers by integer polynomials, where it is conjectured that a separable polynomial f(x) with integer coefficients takes infinitely many squarefree values, barring some simple exceptional cases, in fact that the integers n for which f(n) is squarefree have a positive density. We show that if f(x) in F[t][x] is separable, with square-free content, of bounded degree and height, then…
Given a rational elliptic surface X over an algebraically closed field, we investigate whether a given natural number k can be the intersection number of two sections of X. If not, we say that k a gap number. We try to answer when gap numbers exist, how they are distributed and how to identify them. We use Mordell-Weil lattices as our main tool, which connects the investigation to the classical problem of representing integers by positive-definite quadratic forms.
We show that a rational function $f$ of degree $>1$ on the projective line over an algebraically closed field that is complete with respect to a non-trivial and non-archimedean absolute value has no potentially good reductions if and only if the Berkovich Julia set of $f$ is uniformly perfect. As an application, a uniform regularity of the boundary of each Berkovich Fatou component of $f$ is also established.
In this note, we give an elementary proof of the following classical fact. Any positive definite ternary quadratic form over the rational numbers fails to represent infinitely many positive integers. For any ternary quadratic form (positive definite or indefinite), our method constructs certain congruence classes whose elements, up to a square factor, are the only elements not represented over the rational numbers by that form. In the case of a positive definite ternary form, we show that these classes are non-empty. This shows that the minimum number of variables in a positive definite quadra…
Given an elliptic quartic of type $Y^2=f(X)$ representing an elliptic curve of positive rank over $\Q$, we investigate the question of when the $Y$-coordinate can be represented by a quadratic form of type $ap^2+bq^2$. In particular, we give examples of equations of surfaces of type $c_0+c_1x+c_2x^2+c_3x^3+c_4x^4=(ap^2+bq^2)^2$, $a,b,c \in \Q$ where we can deduce the existence of infinitely many rational points. We also investigate surfaces of type $Y^2=f(a p^2+b q^2)$ where the polynomial $f$ is of degree $3$.
We analyze the structure of a large class of connected algebraic rings over an algebraically closed field of positive characteristic using Greenberg's perfectization functor. We then give applications to rigidity problems for representations of Chevalley groups, recovering, in particular, a rigidity theorem of Seitz.
The equality between the number of odd spin structures on a Riemann surface of genus g, with $2^g - 1$ being a Mersenne prime, and the even perfect numbers, is an indication that the action of the modular group on the set of spin structures has special properties related to the sequence of perfect numbers. A method for determining whether Mersenne numbers are primes is developed by using a geometrical representation of these numbers. The connection between the non-existence of finite odd perfect numbers and the irrationality of the square root of twice the product of a sequence of repunits is …
