Primitive divisors on twists of the Fermat cubic

雅各布梯、Hardy-Littlewood积分（1918）的几乎线性增量、经典Dirichlet除数和（1849）及其与Fermat-Wiles定理的关系

Jacob's ladders, almost linear increments of the Hardy-Littlewood integral (1918), the classical Dirichet's sum of divisors (1849) and their relationship with the Fermat-Wiles theorem

该研究基于雅各布梯推导Fermat-Wiles定理的新等价形式，核心为由Dirichlet的D(x)函数生成的D-等价形式，为数论相关定理研究提供了新的等价表述方向。

The Uniform Primality Conjecture for the Twisted Fermat Cubic

On the twisted Fermat cubic, an elliptic divisibility sequence arises as the sequence of denominators of the multiples of a single rational point. We prove that the number of prime terms in the sequence is uniformly bounded. When the rational point is the image of another rational point under a certain 3-isogeny, all terms beyond the first fail to be primes.

Perfect powers generated by the twisted Fermat cubic

On the twisted Fermat cubic, an elliptic divisibility sequence arises as the sequence of denominators of the multiples of a single rational point. It is shown that there are finitely many perfect powers in such a sequence whose first term is greater than 1. Moreover, if the first term is divisible by 6 and the generating point is triple another rational point then there are no perfect powers in the sequence except possibly an lth power for some l dividing the order of 2 in the first term.

Sato-Tate distributions of twists of the Fermat and the Klein quartics

We determine the limiting distribution of the normalized Euler factors of an abelian threefold A defined over a number field k when A is geometrically isogenous to the cube of a CM elliptic curve defined over k. As an application, we classify the Sato-Tate distributions of the Jacobians of twists of the Fermat and Klein quartics, obtaining 54 and 23, respectively, and 60 in total. We encounter a new phenomenon not visible in dimensions 1 or 2: the limiting distribution of the normalized Euler factors is not determined by the limiting distributions of their coefficients.

The arithmetic of a twist of the Fermat quartic

We study the arithmetic of the twist of the Fermat quartic defined by $X^4 + Y^4 + Z^4 = 0$ which has no $\mathbb{Q}$-rational point. We calculate the Mordell--Weil group of the Jacobian variety explicilty. We show that the degree $0$ part of the Picard group is a free $\mathbb{Z}/2\mathbb{Z}$-module of rank $2$, whereas the Mordell--Weil group is a free $\mathbb{Z}/2\mathbb{Z}$-module of rank $3$. Thus the relative Brauer group is non-trivial. We also show that this quartic violates the local-global property for linear determinantal representations.

五色超立方体Adinkra与广义费马曲线的雅可比

The five-color hypercube Adinkra and the Jacobian of a generalized Fermat curve

本文研究了五色超立方体Adinkra与广义费马曲线雅可比之间的映射，提出了一种组合算法计算高度函数图像，并给出了生成分歧系数的严格界限。