Jonathan Pila
Let X be the graph in the plane of a pfaffian function f (in the sense of Khovanskii). Suppose X is not algebraic. This note gives an upper bound for the number of rational points on X of height up to X. The bound is uniform in the order and degre of f.
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We prove effective Pila--Wilkie estimates for the number of rational points of bounded height lying on certain surfaces defined by Pfaffian functions. The class of surfaces to which our result applies includes, for instance, graphs of unrestricted Pfaffian functions defined on the plane.
This short note describes the connection between strong minimality of the differential equation satisfied by an complex analytic function and the real and imaginary parts of the function being Pfaffian. This connection combined with a theorem of Freitag and Scanlon (2017) provides the answer to a question of Binyamini and Novikov (2017). We also answer a question of Bianconi (2016). We give what seem to be the first examples of functions which are definable in o-minimal expansions of the reals and are differentially algebraic, but not Pfaffian.
This paper is devoted to conditions defined in terms of the generalized shift operator for a rational number to be representable by certain positive generalizations of $q$-ary expansions.
Under fairly natural assumptions, Huang counted the number of rational points lying close to an arc of a planar curve. He obtained upper and lower bounds of the correct order of magnitude, and conjectured an asymptotic formula. In this note, we establish the conjectured asymptotic formula.
In this short note we confirm the relation between the generalized $abc$-conjecture and the $p$-rationality of number fields. Namely, we prove that given K$/\mathbb{Q}$ a real quadratic extension or an imaginary $S_3$-extension, if the generalized $abc$-conjecture holds in K, then there exist at least $c\,\log X$ prime numbers $p \leq X$ for which K is $p$-rational, here $c$ is some nonzero constant depending on K. The real quadratic case was recently suggested by Böckle-Guiraud-Kalyanswamy-Khare.
A note on extended genus fields of Kummer extensions of rational function fields
本文将 Clement 于 1992 年关于有理函数域素数次循环 Kummer 扩张扩展亏格域的构造推广到一般 Kummer 扩张，并证明其原方法依然适用。
We compute the Galois group of the maximal 2-ramified and complexified pro-2-extension of any 2-rational number field.
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.
In this note we study an analogy between a positive definite quadratic form for elliptic curves over finite fields and a positive definite quadratic form for elliptic curves over the rational number field. A question is posed of which an affirmative answer would imply the analogue of the Riemann hypothesis for elliptic curves over the rational number field.
Bourgain, Konyagin and Shparlinski obtained a lower bound for the size of the product set AB when A and B are sets of positive rational numbers with numerator and denominator less or equal than Q. We extend and slightly improve that lower bound using a different approach.
We compute the Galois group of the maximal 2-ramified pro-2-extension of a 2-rational number field
Let C be the image of a canonical embedding C of the Atkin-Lehner quotient X+0(N) associated to the Fricke involution wN. In this note we exhibit some relations among the rational points of C. For each g = 3 (resp. the first g = 4) curve C we found that there are one or more lines (resp. planes) in projectie space whose intersection with C consists entirely of rational Heegner points or the cusp point, where N is prime. We also discuss an explanation of the first non-hyperelliptic exceptional rational point.
In this note, we prove that for every two positive integers $m \geq n \geq 9$, there exist $n$ positive rational numbers whose product is 1 and sum is $m$.
In this note we give an algorithm to explicitly construct the modular parametrization of an elliptic curve over the rationals given the Weierstrass function $\wp (z)$.
We study the field isomorphism problem of cubic generic polynomial $X^3+sX+s$ over the field of rational numbers with the specialization of the parameter $s$ to nonzero rational integers $m$ via primitive solutions to the family of cubic Thue equations $x^3-2mx^2y-9mxy^2-m(2m+27)y^3=λ$ where $λ^2$ is a divisor of $m^3(4m+27)^5$.
In this note we show that the system of equations t_{x}+t_{y}=t_{p},\quad t_{y}+t_{z}=t_{q},\quad t_{x}+t_{z}=t_{r}, where $t_{x}=x(x+1)/2$ is a triangular number, has infinitely many solutions in integers. Moreover we show that this system has rational three-parametric solution. Using this result we show that the system t_{x}+t_{y}=t_{p},\quad t_{y}+t_{z}=t_{q},\quad t_{x}+t_{z}=t_{r},\quad t_{x}+t_{y}+t_{z}=t_{s} has infinitely many rational two-parametric solutions.
In this note we formulate some questions in the study of approximations of reals by rationals of the form a/b^2 arising in theory of Shr"odinger equations. We hope to attract attention of specialists to this natural subject of number theory.
This note describes continued fraction representations for the rational approximations to the zeta function recently found by the author. It is tempting to think that these continued fractions might be analysed using a souped up version of the Worpitzky argument so as to produce zero-free regions for the approximations.
A note on the second supplementary law of rational power residue symbols
本文针对具有特殊和式形式的素数 $p$，利用负指标多对数函数给出 $q$ 次幂剩余符号 $(q/p)_q=1$ 的显式条件，从而推广第二补律。
In this note, basing on a certain functional equation of the dilogarithm function, we establish nontrivial lower bounds for the $p$-adic valuation (where $p$ is a given prime number) of some type of rational numbers involving harmonic numbers. Then we use our estimate to derive the integrality of some sequences of rational numbers, which cannot be seen directly from their definitions.
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 quadratic 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. Recently two sets of formulas were derived providing two solutions for the cuboid factor equations. These two solutions are studied in the present paper. They are proved to coincide with each other …
A note on Bremner's conjecture and uniformity
本文通过高度统一的Mordell定理，作者给出了Bremner猜想中关于有理数域上椭圆曲线的有理点序列的长度与秩之间的关系的更直接证明，并扩展了这些方法到有限生成乘法群和几何等比数列的应用。
A Note On Transcendental Analytic Functions With Rational Coefficients Mapping $\mathbb{Q}$ Into Itself
本文研究将有理数集映射到自身的超越整函数问题，证明了对于任意 $t>0$，不存在分母增长满足 $O(q^{t})$ 且将 $\mathbb{Q}$ 映射到 $\mathbb{Q}$ 的超越整函数 $f\in\mathbb{Q}[[z]]$。
Park City lecture notes: around the inverse Galois problem
本讲义围绕逆伽罗瓦问题，介绍其相关研究，包括诺特问题、刚性方法及单有理簇的算术，基于2022年帕克城讲座。
Rational points on varieties and the Brauer-Manin obstruction
本讲义介绍了有理点存在性的Brauer-Manin障碍，重点探讨理论与计算之间的相互作用。
Notes from lectures given at the Autumn School on Algebraic and Arithmetic Geometry at the Johannes Gutenberg-Universität Mainz in October 2017.
In this note we study some sequences whose ratio converges to the square root of rationals. Further we analyze some related sequences obtained when the above mentioned ratio simplifies.
We consider the polynomials $\displaystyle f(x)=x^d+c$, where $d\ge 2$ and $c\in\mathbb Q$. It is conjectured that if $d=2$, then $f$ has no rational periodic point of exact period $N\ge 4$. In this note, fixing some integer $d\ge 2$, we show that the density of such polynomials with a rational periodic point of any period among all polynomials $f(x)=x^d+c$, $c\in\Q$, is zero. Furthermore, we establish the connection between polynomials $f$ with periodic points and two arithmetic sequences. This yields necessary conditions that must be satisfied by $c$ and $d$ in order for the polynomial $f$ t…
In this note, we show that each positive rational number can be written as $φ(m^2)/φ(n^2)$, where $φ$ is Euler's totient function and $m$ and $n$ are positive integers.
These are the substantially expanded notes of the lectures of JK at the summer school "Higher-Dimensional Geometry over Finite Fields" in Göttingen, June 2007. The first part gives an overview of the methods. The main new result is the construction of rational curves passing through a given collection of points on smooth cubic hypersurfaces over finite fields.
Moderate rank jumps on rational elliptic surfaces via construction of conics
本文通过构造圆锥方法，在有理椭圆面中实现秩的适度跳跃，研究具有高秩纤维的椭圆面家族。
Rational points over C1 fields
本文简要评述 Manin 1966 年关于完美域上有理曲面的论文及其猜想和后续发展，并给出相关背景与研究文献。
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