Yuri Bilu, Alvanos Paraskevas, Poulakis Dimitrios
A classical theorem of Siegel asserts that the set of S-integral points of an algebraic curve C over a number field is finite unless C has genus 0 and at most two points at infinity. In this paper we give necessary and sufficient conditions for C to have infinitely many S-integral points.
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Notes on Algebraic Properties and Non-Standard Analysis of the Ring of Integers Modulo Infinitely Large Primes
本文探讨无限大素数模整数环的代数与模型论性质，结合非标准分析方法扩展超越数理论中的判别准则。
Algebraic functions with infinitely many values in a number field
本文刻画了定义在代数闭包上的代数曲线，其满足存在某数域及其中无限子集，使得对应多项式根均属于该数域。
Suppose that $\al>1$ is an algebraic number and $ξ>0$ is a real number. We prove that the sequence of fractional parts $\{ξ\al^n\},$ $n =1,2,3,...,$ has infinitely many limit points except when $\al$ is a PV-number and $ξ\in \Q(\al).$ For $ξ=1$ and $\al$ being a rational non-integer number, this result was proved by Vijayaraghavan.
We use a generalization of a construction by Ziegler to show that for any field $F$ and any countable collection of countable subsets $A_i \subseteq F, i \in \calI \subset \Z_{>0}$ there exist infinitely many fields $K$ of arbitrary positive transcendence degree over $F$ and of infinite algebraic degree such that each $A_i$ is first-order definable over $K$. We also use the construction to show that many infinitely axiomatizable theories of fields which are not compatible with the theory of algebraically closed fields are finitely hereditarily undecidable.
In this paper, we define $X$-base Fibonacci-Wieferich prime which is a generalized Wieferich prime where $X$ is a finite set of algebraic numbers. We are going to show that there are infinitely many non-$X$-base Fibonacci-Wieferich primes assuming the $abc$-conjecture of Masser-Oesterlé-Szpiro for number fields. We also provide a new conjecture concerning the rank of free part of abelian group generated by all elements in $X$, and we will use the arithmetic point of view and geometric point of view to give heuristic.
Given a holomorphic $C_2$-cofinite vertex operator algebra $V$ with graded dimension $j-744$, Borcherds's proof of the monstrous moonshine conjecture implies any finite order automorphism of $V$ has graded trace given by a "completely replicable function", and by work of Cummins and Gannon, these functions are principal moduli of genus zero modular groups. The action of the monster simple group on the monster vertex operator algebra produces 171 such functions, known as the monstrous moonshine functions. We show that 154 of the 157 non-monstrous completely replicable functions cannot possibly …
On non-holonomicity, transcendence and $p$-adic valuations
本文研究了 $p$-进赋值相关幂级数的性质，证明其在特征 0 下非完整且非代数，找到了使级数值超越的无穷多个有理数与代数无理数，并应用于 $p$-自动序列。
Triple intersections for algebraic curves on the real torus
本文受Corvaja和Zannier问题启发，研究实环面上三条实代数曲线投影的三重交点，对三条直线给出完整分类，对高次曲线用Siegel和Levin定理得有限性结果，并构造无穷多三重交点的显式例子。
We show, assuming Schanuel's conjecture, that every irreducible complex polynomial in two variables where both variables appear has infinitely many algebraically independent solutions of the form (z,e^z).
Let $Γ\subset \overline{\mathbb Q}^{\times}$ be a finitely generated multiplicative group of algebraic numbers. Let $δ, β\in\overline{\mathbb Q}^\times$ be algebraic numbers with $β$ irrational. In this paper, we prove that there exist only finitely many triples $(u, q, p)\inΓ\times\mathbb{Z}^2$ with $d = [\mathbb{Q}(u):\mathbb{Q}]$ such that $$ 0<|δqu+β-p|<\frac{1}{H^\varepsilon(u)q^{d+\varepsilon}}, $$ where $H(u)$ denotes the absolute Weil height. As an application of this result, we also prove a transcendence result, which states as follows: Let $α>1$ be a real number. Let $β$ be an algebr…
We consider the issue of when the L-polynomial of one curve over $\F_q$ divides the L-polynomial of another curve. We prove a theorem which shows that divisibility follows from a hypothesis that two curves have the same number of points over infinitely many extensions of a certain type, and one other assumption. We also present an application to a family of curves arising from a conjecture about exponential sums. We make our own conjecture about L-polynomials, and prove that this is equivalent to the exponential sums conjecture.
We study the algebra $\mathcal{I}^{QM}$ of iterated integrals of quasimodular forms for $\operatorname{SL}_2(\mathbb{Z})$, which is the smallest extension of the algebra $QM_{\ast}$ of quasimodular forms, which is closed under integration. We prove that $\mathcal{I}^{QM}$ is a polynomial algebra in infinitely many variables, given by Lyndon words on certain monomials in Eisenstein series. We also prove an analogous result for the $M_{\ast}$-subalgebra $\mathcal{I}^{M}$ of $\mathcal{I}^{QM}$ of iterated integrals of modular forms.
Given any positive integer n, we prove the existence of infinitely many right triangles with area n and side lengths in certain number fields. This generalizes the famous congruent number problem. The proof allows the explicit construction of these triangles; for this purpose we find for any positive integer n an explicit cubic number field Q(λ) (depending on n) and an explicit point P_λof infinite order in the Mordell-Weil group of the elliptic curve Y^2=X^3-n^2*X over Q(λ).
A sequence of rational points on an algebraic planar curve is said to form an $r$-geometric progression sequence if either the abscissae or the ordinates of these points form a geometric progression sequence with ratio $r$. In this work, we prove the existence of infinitely many rational numbers $r$ such that for each $r$ there exist infinitely many $r$-geometric progression sequences on the unit circle $x^2 + y^2 = 1$ of length at least $3$.
In this paper we show that a polynomial equation admits infinitely many prime-tuple solutions assuming only that the equation satisfies suitable local conditions and the polynomial is sufficiently non-degenerate algebraically. Our notion of algebraic non-degeneracy is related to the $h$-invariant introduced by W. M. Schmidt. Our results prove a conjecture of B. Cook and Á. Magyar \cite{CM} for hypersurfaces of degrees $2$ and $3$.
In the present paper we show that there exist infinitely many consecutive square-free numbers of the form $[αn]$, $[αn]+1$, where $α>1$ is irrational number with bounded partial quotient or irrational algebraic number.
Unlike the real case, there are not many studies and general techniques for providing simultaneous approximations in the field of $p$--adic numbers $\mathbb Q_p$. Here, we study the use of multidimensional continued fractions (MCFs) in this context. MCFs were introduced in $\mathbb R$ by Jacobi and Perron as a generalization of continued fractions and they have been recently defined also in $\mathbb Q_p$. We focus on the dimension two and study the quality of the simultaneous approximation to two $p$-adic numbers provided by $p$-adic MCFs, where $p$ is an odd prime. Moreover, given algebraical…
In general, not much is known about the arithmetic of K3 surfaces. Once the geometric Picard number, which is the rank of the Neron-Severi group over an algebraic closure of the base field, is high enough, more structure is known and more can be said. However, until recently not a single K3 surface was known to have geometric Picard number one. We give explicit examples of such surfaces over the rational numbers. This solves an old problem that has been attributed to Mumford. The examples we give also contain infinitely many rational points, thereby answering a question of Swinnerton-Dyer and …
The abc conjecture implies infinitely many non-Wieferich places for fixed bases in number fields
本文在abc猜想假设下，将非Wieferich素数增长结果推广至数域，证明Ding的结果在虚二次域整数环中除31个例外外对所有基成立。
We prove the following propositions. Theorem 1: Let $M$ be a subfield of a fixed algebraic closure $\tilde \Q$ of $\Q$ whose existential elementary theory is decidable (resp. primitively decidable). Then, M is conjugate to a recursive (resp. primitive recursive) subfield $L \subset \tilde \Q$. Theorem 2: For each positive integer $e$ there are infinitely many $e$-tuples $\boldsymbol σ\in \Gal(\Q)^e$ such that the field $\tilde \Q( {\boldsymbol σ})$ -- the fixed field of $\boldsymbol σ$, is recursive in $\tilde\Q$ and its elementary theory is decidable. Moreover, $\tilde \Q(\boldsymbol σ)$ is P…
Hasse principle violation for algebraic families of del Pezzo surfaces of degree 4 and hyperelliptic curves of genus congruent to 1 modulo 4
本文构造了无限多个显式单参数代数族，包含次数为4的del Pezzo曲面和亏格模4余1的超椭圆曲线，证明其每个K成员都违反Hasse原则，并由Brauer-Manin障碍解释。
Around Fekete's theorem
该研究探讨了复平面上紧致集与代数整数序列的关系，提出了一种在保持高度界限下使共轭值接近集的构造方法，并分析了极限分布的性质。
We generalize Siegel's theorem on integral points on affine curves to integral points of bounded degree, giving a complete characterization of affine curves with infinitely many integral points of degree d or less over some number field. Generalizing Picard's theorem, we prove an analogous result characterizing complex affine curves admitting a nonconstant holomorphic map from a degree d (or less) analytic cover of $\mathbb{C}$.
Infinitely many holes in connectedness loci for collinear affine iterated function systems
研究等距平移共线仿射迭代函数系统连通性轨迹$M_n$的拓扑，证明对每个整数$n\\\ge2$，$M_n$包含无穷多个洞，并通过有限逆树证书和重正化点$\\xi_n$的收敛性严格证明。
Irreducibility of polarized automorphic Galois representations in infinitely many dimensions
本文研究了极化自守伽罗瓦表示在无限多个维度中的不可约性，证明在特定条件下存在一个素数集合，使得对应的伽罗瓦表示在这些素数下都是不可约的。
We introduce an algebraicity criteria. It has the following form: under certain conditions, an analytic subvariety of some algebriac variety over a global field $K$, if it contains many $K$-points, then it is algebraic over $K.$ This gives a way to show the transcendence of points via the transcendence of analytic subvarieties. Such a situation often appears when we have a dynamical system, because we can often produce infinitely many points from one point via iterates. Combing this criteria and the study of invariant subvarieties, we get some results on the transcendence in arithmetic dynamic…
We describe all special curves in the parameter space of complex cubic polynomials, that is all algebraic irreducible curves containing infinitely many post-critically finite polynomials. This solves in a strong form a conjecture by Baker and DeMarco for cubic polynomials. We also prove that an irreducible component of the algebraic curve consisting of those cubic polynomials that admit an orbit of any given period and given multiplier is special if and only if the multiplier is 0.
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