On Some Diophantine Parameters of the Cyclic Torsion Subgroups of Odd Order of Elliptic Curves over $\mathbb{Q}$
Derong Qiu
本文研究有理数域上椭圆曲线奇阶循环挠子群，并给出了其若干显式丢番图参数。
本文给出了有理数域上椭圆曲线奇阶循环挠子群的若干显式丢番图参数。
In this paper, we give some explicit Diophantine parameters of the cyclic torsion subgroups of odd order of elliptic curves over $\mathbb{Q}$.
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本文研究正则丢番图三元组诱导的椭圆曲线的算术，明确其挠子群结构，给出二次域上获3阶点的判定准则并验证特定族无此情况，还分析另一类D(-k²)三元组诱导曲线的挠与秩。
We prove asymptotic formulas for cyclicity of reductions of elliptic curves over the rationals in a family of curves having specified torsion. These results agree with established conditional results and with average results taken over larger families. As a key tool, we prove an analogue of a result of Vlăduţ that estimates the number of elliptic curves over a finite field with a specified torsion point and cyclic group structure.
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Non-cyclic Torsion of Elliptic Curves Over Imaginary Quadratic Fields of Class Number 1
本文研究类数1非分圆虚二次域上具有全2-挠点的椭圆曲线，确定了其在基域二次扩张上的挠群所有可能结构。
Let $p$ be a prime integer, $\mathbb{Z}_p$ the finite field of order $p$ and $\mathbb{Z}^{*}_{p}$ is its multiplicative cyclic group. We consider the Diophantine equation $x^n + y^n = z^n$ with $1 \leq n \leq \frac{p - 1}{2}$. Our main aim in this paper is to give optimal conditions or relationships between the exponent $n$ and the prime $p$ to determine the existence of nontrivial solutions of the diophantine equation $x^n + y^n = z^n$ with $1 \leq n \leq p -1 $, in finite fields $\mathbb{Z}_p$.
A positive density of elliptic curves are diophantine stable in certain Galois extensions
本文证明，对于循环p-扩张L/Q（p=3或5），存在正密度的椭圆曲线E（按高度排序）在L中具有丢番图稳定性。
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本文研究有理数域椭圆曲线在素数幂循环扩张中的Mordell-Weil群与Tate-Shafarevich群稳定性，提出“Sha”-稳定性概念，并利用算术统计与岩泽理论证明非CM椭圆曲线在几乎所有素数处稳定。
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Solution of certain Diophantine equations in Gaussian integers
针对满足特定同余和勒让德符号条件的素数 p、q，论文利用两族椭圆曲线的挠点与 Q(i) 上的秩，证明两类四次高斯整数丢番图方程仅有平凡解。
Endomorphism algebras of abelian varieties with large cyclic 2-torsion field over a given field
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本文将代数环面上挠点的伽罗瓦均匀分布推广到奇异检验函数，得到带幂节省的定量衰减估计，并据此证明了一类非环面除子的 Ih 整性有限性猜想。
The classification of elliptic curves E over the rationals Q is studied according to their torsion subgroups E_{tors}(Q) of rational points. Explicit criteria for the classification are given when E_{tors}(Q) are cyclic groups with even orders. The generator points P of E_{tors}(Q) are also explicitly presented in each case. These results, together with recent results of K. Ono, completely solve the problem of the mentioned explicit classification when E has a rational point of order 2.
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This paper presents a new result concerning the distribution of 2-Selmer ranks in the quadratic twist family of an elliptic curve over an arbitrary number field K with a single point of order two that does not have a cyclic 4-isogeny defined over its two-division field. We prove that at least half of all the quadratic twists of such an elliptic curve have arbitrarily large 2-Selmer rank, showing that the distribution of 2-Selmer ranks in the quadratic twist family of such an elliptic curve differs from the distribution of 2-Selmer ranks in the quadratic twist family of an elliptic curve having…
This paper presents a new result concerning the distribution of 2-Selmer ranks in the quadratic twist family of an elliptic curve with a single point of order two that does not have a cyclic 4-isogeny defined over its two-division field. We prove that at least half of all the quadratic twists of such an elliptic curve have arbitrarily large 2-Selmer rank, showing that the distribution of 2-Selmer ranks in the quadratic twist family of such an elliptic curve differs from the distribution of 2-Selmer ranks in the quadratic twist family of an elliptic curve having either no rational two-torsion o…
This paper considers solutions (x_1, x_2, ..., x_n) to the cyclic system of n simultaneous congruences r (x_1x_2 ...x_n)/x_i = s (mod |x_i|), for fixed nonzero integers r,s with r>0 and gcd(r,s)=1. It shows this system has a finite number of solutions in positive integers x_i >1 having gcd(x_1x_2...x_n, s)=1, obtaining a sharp upper bound on the maximal size of the solutions in many cases. This bound grows doubly-exponentially in n. It shows there are infinitely many such solutions when the positivity restriction is dropped, when r=1, and not otherwise. The problem is reduced to the study of i…
Completing the classification of torsion subgroups for rational elliptic curves over sextic fields
该研究证明了有理椭圆曲线在六次域上不存在 C_3 ⊕ C_{18} torsion 子群，通过 Galois 表示理论和 Diophantine 几何方法完成分类。
Elliptic $K$-curves are elliptic curves defined over some field extension $L/K$ that are isogenous to all of their Galois conjugates. We present a new result on $K$-curves $E$ that are given by a $K$-rational orbit $\{τ, -1/Nτ\}$ of the Fricke involution on $Y_0(N)$, giving a simple Diophantine condition on the extension $L/K$ that determines which twists of $E$ allow the isogeny between Galois conjugates to be defined over $L$. To support and illustrate this result, we also discuss parameterizations of cyclic $N$-isogenies corresponding to points on modular curves $X_0(N)$ of genus $0$. These…
Weighted Fruit Diophantine Equations and Hyperelliptic Curves
研究加权水果丢番图方程在特定参数条件下的整数解不存在性，通过关联超椭圆曲线将解的不存在性转化为有理扭点的结果。
Skew cyclic codes through Drinfeld modules
本文利用超奇异 Drinfeld 模的挠元构造秩度量下的斜循环码，证明码与子模的格反同构及秩-BCH 界，设计距离等于真实距离。
Let $\mathcal{E}$ be a $\mathbb{Q}$-isogeny class of elliptic curves defined over $\mathbb{Q}$. The isogeny graph associated to $\mathcal{E}$ is a graph which has a vertex for each elliptic curve in the $\mathbb{Q}$-isogeny class $\mathcal{E}$, and an edge for each cyclic $\mathbb{Q}$-isogeny of prime degree between elliptic curves in the isogeny class, with the degree recorded as a label of the edge. In this paper, we define an isogeny-torsion graph to be an isogeny graph where, in addition, we label each vertex with the abstract group structure of the torsion subgroup over $\mathbb{Q}$ of th…
In this paper, we construct an infinite family of elliptic curves whose rank is exactly two and the torsion subgroup is a cyclic group of order two or three, under the parity conjecture.
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