Magdaléna Tinková
This paper focuses on greedy expansions, one possible representation of numbers, and on arithmetical operations with them. Performing addition or multiplication some additional digits can appear. We study bounds on the number of such digits assuming the finiteness of the expansion of the considered sum or product, especially for the case of cubic Pisot units.
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Euler showed that there can be no more than three integer squares in arithmetic progression. In quadratic number fields, Xarles has shown that there can be arithmetic progressions of five squares, but not of six. Here, we prove that there are no cubic number fields which contain five squares in arithmetic progression.
Arithmetic purity of strong approximation for complete intersections of large dimension
该研究对数域上光滑几何整的三次超曲面，建立了强逼近的算术纯性，给出了一般情况与有理数域下不同的维度阈值。
The divisor function along arithmetic progressions and binary cubic polynomials
该研究提出带两个小因子模的算术级数中除数函数的新等分布估计，给出两项应用：指数2/3的几乎所有模的算术级数上除数函数的渐近公式，以及沿非齐次二元三次多项式XY²+1的除数函数渐近公式。
A cubic (resp. biquadratic) theta series is a power series whose n-th coefficient is equal to 1 if n is a perfect cube (resp. fourth power) and zero otherwise. We improve on a result of Bradshaw by showing that such series is not a cubic (resp. biquadratic) algebraic number when evaluated at reciprocals of integers. The proof relies on a "nested gaps technique" for linear independence and on recent results by the author on Waring's problem for cubes and biquadrates.
Let $b(n)$ denote the number of cubic partition pairs of $n$. We give affirmative answer to a conjecture of Lin, namely, we prove that $$b(49n+37)\equiv 0 \pmod{49}.$$ We also prove two congruences modulo $256$ satisfied by $\overline{b}(n)$, the number of overcubic partition pairs of $n$. Let $\overline{a}(n)$ denote the number of overcubic partition of $n$. For a fixed positive integer $k$, we further show that $\overline{b}(n)$ and $\overline{a}(n)$ are divisible by $2^k$ for almost all $n$. We use arithmetic properties of modular forms to prove our results.
Arithmetic Properties of Generalized Cubic and Overcubic Partitions
本文研究了广义立方和超立方划分函数的若干同余关系，利用q级数技术和模形式理论证明了模2和12的幂次同余关系及密度结果。
Arithmetic of cubic number fields: Jacobi-Perron, Pythagoras, and indecomposables
本文建立 Jacobi–Perron 等多维连分数与全实三次数域加法不可分解整数的联系，刻画 Ennola 族不可分解元与 Pythagoras 数，并计算比较 Shanks 族中的周期 JPA 展开。
Let $b(n)$ denote the number of cubic partition pairs of $n$. In this paper, we aim to provide a strategy to obtain arithmetic properties of $b(n)$. This gives affirmative answers to two of Lin's conjectures.
Cubic arithmetic-geometric mean and isogenies of Hesse elliptic curves
本文在特征非三的任意域上构造Hesse椭圆曲线的三次同源，对应三次算术几何平均的一步，并应用于特征三中普通椭圆曲线的Serre-Tate典范提升，同时探讨与超几何函数的联系。
One of the main themes in this thesis is the description of the signature of both the infinite place and the finite places in cubic function fields of any characteristic and quartic function fields of characteristic at least 5. For these purposes, we provide a new theory which can be applied to cubic and quartic function fields and to even higher dimensional function fields. One of the striking advantages of this theory to other existing methods is that is does not use the concept of p-adic completions and we can dispense of Cardano's formulae. Another key result comprises the construction of …
In this paper we compute the distributions of various markings on smooth cubic surfaces defined over the finite field $\mathbb{F}_q$, for example the distribution of pairs of points, `tritangents' or `double sixes'. We also compute the (rational) cohomology of certain associated bundles and covers over complex numbers.
We show that binary forms of degree n less than seven parameterize rings, generalizing a result of Levi on binary cubic forms parameterizing cubic rings, which can be related to results of Bhargava. The question of whether or not a binary form of degree n parameterizes a ring of rank $n$ depends on a symbolic calculation with matrices.
In the present paper, we will show that three apparently disjoint objects: Galois representations arising from twenty-seven lines on a cubic surface (number theory and arithmetic algebraic geometry), Picard modular forms (automorphic forms), rigid Calabi-Yau threefolds and their arithmetic (Diophantine geometry) are intimately related to Hessian polyhedra and their invariants. We construct a Galois representation whose image is a proper subgroup of $W(E_6)$, the Weyl group of the exceptional Lie algebra $E_6$. We give a conjecture about the identification of two different kinds of $L$-function…
Arithmetic Period Map and Complex Multiplication for Cubic Fourfolds
本文构造了三次四次曲面的算术周期映射，并发展了其复乘法理论，证明了CM类型的三次四次曲面定义在阿贝尔扩张上。
Arithmetic properties for generalized cubic partitions and overpartitions modulo a prime
本文引入广义三次分拆概念，用初等（函数方程）和模形式两种方法证明了若干类似拉马努金型的新同余式，并得到广义超三次分拆的相关类似结果，丰富了分拆数论的算术性质研究。
We study dynamical systems induced by birational automorphisms on smooth cubic surfaces defined over a number field $K$. In particular we are interested in the product of non-commuting birational Geiser involutions of the cubic surface. We present results describing the sets of $K$ and $\bar{K}$-periodic points of the system, and give a necessary and sufficient condition for a dynamical local-global property called strong residual periodicity. Finally, we give a dynamical result relating to the Mordell--Weil problem on cubic surfaces.
Let $K$ be a number field, which is tame and non totally real. In this article we give a numerical criterion, depending only on the ramification behavior of ramified primes in $K$, to decide whether or not the integral trace of $K$ is isometric to the integral trace of another number field $L$. As a byproduct of our proofs here, and in contrast with our previous results for cubic fields of positive discriminant, we show that for cubic fields of negative discriminant isometry between integral traces is equivalent to equality of discriminants.
It is shown that $c=-29/16$ is the unique rational number of smallest denominator, and the unique rational number of smallest numerator, for which the map $f_c(x) = x^2+c$ has a rational periodic point of period $3$. Several arithmetic conditions on the set of all such rational numbers $c$ and the rational orbits of $f_c(x)$ are proved. A graph on the numerators of the rational $3$-periodic points of maps $f_c$ is considered which reflects connections between solutions of norm equations from the cubic field of discriminant $-23$.
Following a strategy suggested by Michel--Venkatesh, we study the cubic moment of automorphic $L$-functions on $\operatorname{PGL}_2$ using regularized diagonal periods of products of Eisenstein series. Our main innovation is to produce vectors whose integral transforms achieve arbitrarily weighted moments. Applications include general Motohashi-type identities and Weyl-type subconvex bounds for some families of $L$-functions, extending some results of Conrey--Iwaniec and Petrow--Young to the number field setting. We deduce improved estimates for representation numbers of ternary quadratic for…
The geometry of secondary terms in arithmetic statistics
本文证明了函数域三次扩张计数中次级项的存在，利用几何参数化方法对光滑曲线计数，推广了有理数域上的结果。
Counting polynomials with positive roots
本文研究所有根均为正实数的首一整系数多项式的计数问题，通过估计满足麦克劳林不等式的整数序列建立固定迹的渐近公式，并对三次多项式给出更精确结果，还证明正比例的三次多项式具有无平方因子判别式。
The number of non-isomorphic cubic fields L sharing a common discriminant d(L) = d is called the multiplicity m = m(d) of d. For an assigned value of d, these fields are collected in a multiplet M(d) = (L(1) ,..., L(m)). In this paper, the information in all existing tables of totally real cubic number fields L with positive discriminants d(L) < 10000000 is extended by computing the differential principal factorization types tau(L) in (alpha1, alpha2, alpha3, beta1, beta2, gamma, delta1, delta2, epsilon) of the members L of each multiplet M(d) of non-cyclic fields, a new kind of arithmetical i…
For an infinite family of monogenic trinomials $P(X) = X^3\pm 3rbX-b$ in $\mathbb{Z}\lbrack X\rbrack$, arithmetical invariants of the cubic number field $L = \mathbb{Q}(θ)$, generated by a zero $θ$ of $P(X)$, and of its Galois closure $N = L(\sqrt{d(L)})$ are determined. The conductor $f$ of the cyclic cubic relative extension $N/K$, where $K = \mathbb{Q}(\sqrt{d(L)})$ denotes the unique quadratic subfield of $N$, is proved to be of the form $3^eb$ with $e\in\lbrace 1,2\rbrace$, which admits statements concerning primitive ambiguous principal ideals, lattice minima, and independent units in $L…
Heavenly Elliptic Curves over Cubic Number Fields
本文研究三次数域上的天椭圆曲线，引入完全平衡概念，证明其挠表示迹非满射，给出平衡性素数界，并与CM曲线比较。
Two-term dilogarithms, mixed-base Nahm sums, and Fricke symmetry
本文研究两项Rogers二对数关系对秩二Nahm系统的算术约束，确定特定族中的允许步长，证明指标13系统的两项恒等式并精确再现Fricke振幅，同时指出相关乘积求值仍为猜想。
Exceptional covers of the projective line with genus-one Galois closure: arithmetic forms over finite fields
本文对几何伽罗瓦闭包亏格1的可分不可约驯服例外射影直线映射分类，给出有限域上算术固定域形式的编码准则，获秩2支撑公式等结果，明确特征与障碍的相关性质。
Arithmetic aspects of the Burkhardt quartic threefold
本文研究 Burkhardt 四次三维簇的算术性质，证明其在特征非 3 域上有理、计算有限域 zeta 函数，并显式给出 genus 2 泛曲线模型及 j-平面与 3 阶子群、Hesse 铅笔的关系。
Geometric families of multiple elliptic Gamma functions and arithmetic applications, III
本文作为高阶椭圆伽马函数算术应用系列第三篇，升级了前人的椭圆单位构造，推测仅含一个复 place 的数域上的高阶椭圆单位为代数单位并满足克罗内克极限公式，且通过多次数域示例验证该推测。
Counting integral points on symmetric varieties with applications to arithmetic statistics
本文结合 Bhargava 几何数论与动力点计数方法，提出对称簇整点计数新技术，并应用于三次数域类群 2-挠子群和椭圆曲线 2-Selmer 群的薄族分布，揭示了小范数生成元对平均大小的增大效应。
In this paper we compare several properties and constructions of the Carlitz polynomials and digit derivatives for continuous functions on $\F_q[[T]].$ In particular, we show a close relation between them as orthonormal bases. Moreover, parallel to Carlitz's coefficient formula, we give the closed formula for the expansion coefficients in terms of the digit derivatives.
Simultaneous Niven Numbers in Arithmetic Progressions for Power-Related Bases
本文研究了在幂相关底数下，等差数列中同时满足多个底数尼文数的性质，通过稀疏重复数构造证明了相关结论。
We are interested in extending normal bases of $\mathbf{F}_{\!2^n}/\mathbf{F}_{\!2}$ to bases of $\mathbf{F}_{\!2^{nd}}/\mathbf{F}_{\!2}$ which allow fast arithmetic in $\mathbf{F}_{\!2^{nd}}$. This question has been recently studied by Thomson and Weir in case $d$ is equal to $2$. We construct efficient extended bases in case $d$ is equal to $3$ and $4$. We also give conditions under which Thomson-Weir construction can be combined with ours.
The different for base change of arithmetic curves
本文提出一种通过同时骨架和骨架Riemann-Hurwitz公式研究算术曲线约化类型及狂分歧基变换的方法，给出新证明并解决Lorenzini关于欧拉示性数的问题。
What is the distance between 11 (a prime number) and 12 (a highly composite number)? If your answer is 1, then ask yourself "is this reasonable?" In this work, we will introduce a distance between natural numbers based on their arithmetic properties, instead of their position on the real line.
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