Claude Levesque, Michel Waldschmidt
To each non totally real cubic extension $K$ of $\Q$ and to each generator $α$ of the cubic field $K$, we attach a family of cubic Thue equations, indexed by the units of $K$, and we prove that this family of cubic Thue equations has only a finite number of integer solutions, by giving an effective upper bound for these solutions.
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In this paper, we investigate the common index divisors of cyclic cubic fields. Let $a,b,c,d$ and $k$ are integers, we then solve the following Thue cubic equations:: \[ax^3+bx^2y+cxy^2+dy^3= k\ \] when $a,bc+d$ are odd and $3$ doesn't divide $v_2(k)$.
We revisit a work by R. Okazaki and prove that for every cubic binary form F(x, y) with large enough discriminant, the Thue equation |F(x, y)| = 1 has at most 7 solutions in integers x and y.
We exactly determine the integral solutions to a previously untreated infinite family of cubic Thue equations of the form $F(x,y)=1$ with at least $5$ such solutions. Our approach combines elementary arguments, with lower bounds for linear forms in logarithms and lattice-basis reduction.
Let $F_n$ denote the $n$-th Fibonacci number and $L_n$ the $n$-th Lucas number. We completely solve the family of cubic Thue equations $${(X-F_nY)(X-L_nY)X-Y^3=\pm1}$$ and show that there are no non-trivial solutions for $n\neq 1,3$.
We will give an explicit upper bound for the number of solutions to cubic inequality |F(x, y)| \leq h, where F(x, y) is a cubic binary form with integer coefficients and positive discriminant D. Our upper bound is independent of h, provided that h is smaller than D^{1/4}.
In this paper we completely solve the family of parametrised Thue equations \[ X(X-F_n Y)(X-2^n Y)-Y^3=\pm 1, \] where $F_n$ is the $n$-th Fibonacci number. In particular, for any integer $n\geq 3$ the Thue equation has only the trivial solutions $(\pm 1,0), (0,\mp 1), \mp(F_n,1), \mp(2^n,1)$.
Let $m\geq -1$ be an integer. We give a correspondence between integer solutions to the parametric family of cubic Thue equations \[ X^3-mX^2Y-(m+3)XY^2-Y^3=λ\] where $λ>0$ is a divisor of $m^2+3m+9$ and isomorphism classes of the simplest cubic fields. By the correspondence and R. Okazaki's result, we determine the exactly 66 non-trivial solutions to the Thue equations for positive divisors $λ$ of $m^2+3m+9$. As a consequence, we obtain another proof of Okazaki's theorem which asserts that the simplest cubic fields are non-isomorphic to each other except for $m=-1,0,1,2,3,5,12,54,66,1259,2389…
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 $m^3(4m+27)^5$.
Let $(A_n)_{n\in \mathbb{N}}, (B_n)_{n\in \mathbb{N}} \in \mathbb{Z}^{\mathbb{N}}$ be two linear-recurrent sequences that meet a dominant root condition and a few more technical requirements. We show that the split family of Thue equations \[ |X(X-A_n Y)(X-B_n Y) - Y^3| = 1 \] has but the trivial solutions $\pm\{ (0,1), (1,0), (A_n,1), (B_n,1) \}$, if the parameter $n$ is larger than some effectively computable constant.
The families of simplest cubic, simplest quartic and simplest sextic fields and the related Thue equations are well known. The family of simplest cubic Thue equations was already studied in the relative case, over imaginary quadratic fields. In the present paper we give a similar extension of simplest quartic and simplest sextic Thue equations over imaginary quadratic fields. We explicitly give the solutions of these infinite parametric families of Thue equations over arbitrary imaginary quadratic fields.
E. Thomas was one of the first to solve an infinite family of Thue equations, when he considered the forms $F_n(X, Y )= X^3 -(n-1)X^2Y -(n+2)XY^2 -Y^3$ and the family of equations $F_n(X, Y )=\pm 1$, $n\in {\mathbf N}$. This family is associated to the family of the simplest cubic fields ${\mathbf Q}(λ)$ of D. Shanks, $λ$ being a root of $F_n(X,1)$. We introduce in this family a second parameter by replacing the roots of the minimal polynomial $F_n(X, 1) $ of $λ$ by the $a$-th powers of the roots and we effectively solve the family of Thue equations that we obtain and which depends now on the …
For any nonzero $h\in\mathbb{Z}$, we prove that a positive proportion of integral binary cubic forms $F$ do locally everywhere represent $h$ but do not globally represent $h$; that is, a positive proportion of cubic Thue equations $F(x,y)=h$ fail the integral Hasse principle. Here, we order all classes of such integral binary cubic forms $F$ by their absolute discriminants. We prove the same result for Thue equations $G(x,y)=h$ of any fixed degree $n \geq 3$, provided that these integral binary $n$-ic forms $G$ are ordered by the maximum of the absolute values of their coefficients.
We show how the theory of linear forms in two logarithms allows one to get effective irrationality measures for $n$-th roots of rational numbers ${a \over b}$, when $a$ is very close to $b$. We give a $p$-adic analogue of this result under the assumption that $a$ is $p$-adically very close to $b$, that is, that a large power of $p$ divides $a-b$. As an application, we solve completely certain families of Thue-Mahler equations. Our results illustrate, admittedly in a very special situation, the strength of the known estimates for linear forms in logarithms.
In the first part we construct algorithms which we apply to solve S-unit, Mordell, cubic Thue, cubic Thue-Mahler and generalized Ramanujan-Nagell equations. As a byproduct we obtain alternative practical approaches for various classical Diophantine problems, including the fundamental problem of finding all elliptic curves over Q with good reduction outside a given finite set of rational primes. To illustrate the utility of our algorithms we determined the solutions of large classes of equations, containing many examples of interest which are out of reach for the known methods. In addition we u…
On split families of Thue equations with linear recurrence sequences as factors
研究以线性递推序列为参数函数的 Thue 方程分裂族，证实了在参数大于某可计算常数时存在显式解集的结论对特定递推序列族同样成立。
Let $α$ be an algebraic number of degree $d\ge 3$ and let $K$ be the algebraic number field $\Q(α)$. When $\varepsilon$ is a unit of $K$ such that $\Q(α\varepsilon)=K$, we consider the irreducible polynomial $f_\varepsilon(X) \in \Z[X]$ such that $f_\varepsilon(α\varepsilon)=0$. Let $F_\varepsilon(X,Y)$ be the irrreducible binary form of degree $d$ associated to $f_{\varepsilon}(X) $ under the condition $F_{\varepsilon}(X,1)=f_{\varepsilon}(X)$. For each positive integer $m$, we want to exhibit an effective upper bound for the solutions $(x,y,\varepsilon)$ of the diophantine inequation $|F_\va…
Let $h(x,y)$ be a non-degenerate binary cubic form with integral coefficients, and let $S$ be an arbitrary finite set of prime numbers. By a classical theorem of Mahler, there are only finitely many pairs of relatively prime integers $x,y$ such that $h(x,y)$ is an $S$-unit. In the present paper, we reverse a well known argument, which seems to go back to Shafarevich, and use the modularity of elliptic curves over $\mathbb{Q}$ to give upper bounds for the number of solutions of such a Thue-Mahler equation. In addition, our methods gives an effective method for determining all solutions, and we …
Twisting a binary form $F_0(X,Y)\in{\mathbb{Z}}[X,Y]$ of degree $d\ge 3$ by powers $\upsilon^a$ ($a\in{\mathbb{Z}}$) of an algebraic unit $\upsilon$ gives rise to a binary form $F_a(X,Y)\in{\mathbb{Z}}[X,Y]$. More precisely, when $K$ is a number field of degree $d$, $σ_1,σ_2,\dots,σ_d$ the embeddings of $K$ into $\mathbb{C}$, $α$ a nonzero element in $K$, $a_0\in{\mathbb{Z}}$, $a_0>0$ and $$ F_0(X,Y)=a_0\displaystyle\prod_{i=1}^d (X-σ_i(α) Y), $$ then for $a\in{\mathbb{Z}}$ we set $$ F_a(X,Y)=\displaystyle a_0\prod_{i=1}^d (X-σ_i(α\upsilon^a) Y). $$ Given $m\ge 0$, our main result is an effect…
Complete Resolution Of A Family Of Twisted Thue Equations
本文求解了Lévesque与Waldschmidt提出的一类扭曲Thue方程，找出其所有整数解，解答了相关公开问题。
Let $K$ be a number field, let $S$ be a finite set of places of $K$ containing the archimedean places and let $μ$, $α_1,α_2,α_3$ be non--zero elements in $K$. Denote by $\OS$ the ring of $S$--integers in $K$ and by $\OS^\times$ the group of $S$--units. Then the set of equivalence classes (namely, up to multiplication by $S$--units) of the solutions $(x,y,z,\varepsilon_1, \varepsilon_2,\varepsilon_3,\varepsilon)\in\OS^3\times(\OS^\times)^4$ of the diophantine equation $$ (X-α_1 E_1 Y) (X-α_2E_2 Y) (X-α_3E_3 Y)Z=μE, $$ satisfying $\Card\{α_1\varepsilon_1,α_2\varepsilon_2,α_3\varepsilon_3\}= 3$, …
We consider the Diophantine equation X^n - 1 = B.Z^n, where B in Z is understood as a parameter. We prove that if the equation has a solution, then either the Euler totient of the radical, phi(rad (B)), has a common divisor with the exponent n, or the exponent is a prime and the solution stems from a solution to the diagonal case of the Nagell-Ljunggren equation: (X^n-1)/(X-1) = n^e.Y^n; e = 0 or 1. This allows us to apply recent results on this equation to the binary Thue equation in question. In particular, we can then display parametrized families for which the Thue equation has no solution…
Power-saving bounds for Thue--Mahler and Mordell equations
该研究针对三次Thue--Mahler方程证明了对regulator具节能依赖的有效界，将Mordell方程整数解的Stark界指数降至1/2+ε，改进了相关分离下界并在Frey高度猜想的导体方面取得节能进展。
We describe a procedure for generating families of cyclic cubic fields with explicit fundamental units. This method generates all known families and gives new ones.
Some upper bounds for the number of monogenizations of quartic orders are established by considering certain classical Diophantine equations, namely index form equations in quartic number fields, and cubic and quartic Thue equations.
Variance of point-counts for families of cubic curves over $\mathbb{F}_p$ and Jacobsthal sums
本文显式计算了单参数三次曲线族上的点数方差，并重点展示了涉及 Jacobsthal 和的方差求值结果。
Lower bound estimates for the rank of universal quadratic forms in some families of real cubic fields with density one
本文在密度为一的实三次域族中，利用不可分解整数和短向量性质，给出了泛二次型秩的明确下界估计，并推广到表示给定整数所有倍数的情形。
Evaluation of eight different families of cubic Euler sums
研究四种、五种和六种三次欧拉和，通过三种不同的分母类型计算非线性欧拉和，利用zeta值和polylogarithmic值明确表达结果。
On the asymptotics of cubic fields ordered by general invariants
该研究引入三次域的广义判别式不变量，推导按其排序的三次域族的渐近性质，并确定满足Malle–Bhargava启发式的族类，丰富了数论中三次域的相关研究。
We study the structure of the codifferent and of additively indecomposable integers in families of totally real cubic fields. We prove that for cubic orders in these fields, the minimal trace of indecomposable integers multiplied by totally positive elements of the codifferent can be arbitrarily large. This is very surprising, as in the so far studied examples of quadratic and simplest cubic fields, this minimum is 1 and 2. We further give sharp upper bounds on the norms of indecomposable integers in our families.
Toroidal families and averages of $L$-functions, II: cubic moments
本文研究了狄利克雷字符下三次矩的 $L$-函数平均，并探讨了其与迹函数双线性形式估计及有限域解个数界的关系。
Using the Thue-Siegel method, we obtain effective improvements on Liouville's irrationality measure for certain one-parameter families of algebraic numbers, defined by equations of the type $(t-a)Q(t)+P(t)=0$. We apply these to some corresponding Diophantine equations. We obtain bounds for the size of solutions, which depend polynomially on $a$, and bounds for the number of these solutions, which are independent of $a$ and in some cases even independent of the degree of the equation.
The cubic moment of $L$-functions for specified local component families
本文研究了 $L$-函数中心值的立方矩在特定局部分量家族上的 Lindelöf 平均上界，通过新发展的 Petersson/Bruggeman-Kuznetsov 公式，推导了 Weyl 强子凸界，尤其在平方满和深度方面。
We present two families of automatic sequences that define algebraic continued fractions in characteristic $2$. The period-doubling sequence belongs to the first first family $\mathcal{P}$; and its sum, the Thue-Morse sequence, belongs to the second family $\mathcal{G}$. The family $\mathcal{G}$ contains all the iterated sums of sequences from the $\mathcal{P}$ and more.
The authors study the central values of L-functions in certain families; in particular they bound the sum of the cubes of these values.Contents:
Recently, using modular forms and Smoot's {\tt Mathematica} implementation of Radu's algorithm for proving partition congruences, Merca proved the following two congruences: For all $n\geq 0,$ \begin{align*} A(9n+5) & \equiv 0 \pmod{3}, \\ A(27n+26) & \equiv 0 \pmod{3}. \end{align*} Here $A(n)$ is closely related to the function which counts the number of {\it cubic partitions}, partitions wherein the even parts are allowed to appear in two different colors. Indeed, $A(n)$ is defined as the difference between the number of cubic partitions of $n$ into an even numbers of parts and the number of…
Large values of cubic and quartic Dirichlet $L$-functions of prime moduli
本文在广义黎曼假设下，应用谐振子方法研究素数模的三次与四次Dirichlet L-函数族，展示其在中心点处的大值。
We show how to construct infinite families of explicitly determined cubic number fields whose class group has a subgroup isomorphic to $(\mathbb{Z}/2)^8$ using degree $1$ del Pezzo surfaces. We illustrate the method and provide an example of such a family.
$K_2$ of families of elliptic curves over non-Abelian cubic and quartic fields
本文给出了非阿贝尔三次和四次域上椭圆曲线族的两种构造，在驯顺符号核中分别获得三个和四个线性无关的“积分”元素，证明了不忽略挠元的新积分性判据，并对90余条曲线数值验证了 Beilinson 猜想。
We study the one-level density for families of L-functions associated with cubic Dirichlet characters defined over the Eisenstein field. We show that the family of $L$-functions associated with the cubic residue symbols $χ_n$ with $n$ square-free and congruent to 1 modulo 9 satisfies the Katz-Sarnak conjecture for all test functions whose Fourier transforms are supported in $(-13/11, 13/11)$, under GRH. This is the first result extending the support outside the \emph{trivial range} $(-1, 1)$ for a family of cubic L-functions. This implies that a positive density of the L-functions associated w…
In this note we study the integer solutions of Cayley's cubic equation. We find infinite families of solutions built from recurrence relations. We use these solutions to solve certain general Pell equations. We also show the similarities and differences to Markov numbers. In particular we introduce new formulae for the solutions to Cayley's cubic equation in analogy with Markov numbers and discuss their distinctions.
Rank distribution in cubic twist families of elliptic curves
本文研究椭圆曲线三次扭族的秩分布，通过伽罗瓦上同调技术证明存在无限多无立方因子扭参数使对应扭曲线的3-Selmer群为零、秩为0。
Continued fractions of cubic Laurent series
本文为$\mathbb{Q}[[t^{-1}]]$中的Laurent级数族构造连分数展开式，应用超几何方法推导三次多项式实根有理逼近下界，并证明三次无理数存在可显式计算的闭形式连分数展开。
Non-vanishing of primitive cubic, quartic and sextic L-functions
该研究在非库默尔情形下，通过适配Gao和Zhao的截断指数方法，改进了三次狄利克雷$L$-函数的非零比例下界，对四次与六次族得到正下界及六次一阶矩渐近公式，所有结果均无条件成立。
On the greatest prime factor of polynomial values and subexponential Szpiro in families
本文结合模方法与对数线性形式，为椭圆曲面上的Szpiro猜想及二次、三次多项式值的最大素因子下界问题获得改进的无条件界，并扩展了先前关于$n^2+1$的结果。
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