论文标题
在全球函数场上椭圆曲线的循环质量扭曲系族的主要selmer等级
On the prime Selmer ranks of cyclic prime twist families of elliptic curves over global function fields
论文作者
论文摘要
修复质数$ p $。令$ \ mathbb {f} _q $为特征性竞争到2、3和$ p $的有限字段，其中还包含unity $μ_p$的原始$ p $ - th。根据Swinnerton-Dyer和Klagsbrun，Mazur和Rubin的作品，我们证明，在$ \ Mathbb {f} _q（t）$限制下，bhar bhar的概率分配了$ \ mathbb {f} $ the bhar的概率，selmer群体大小的概率分布在一个环状椭圆形曲线上的概率分布Lenstra，Poonen和Rains具有明确的误差界限。证明这些结果的关键工具是关于全球函数字段，Erdös-kac定理和马尔可夫链的几何形象的Riemann假设。
Fix a prime number $p$. Let $\mathbb{F}_q$ be a finite field of characteristic coprime to 2, 3, and $p$, which also contains the primitive $p$-th root of unity $μ_p$. Based on the works by Swinnerton-Dyer and Klagsbrun, Mazur, and Rubin, we prove that the probability distribution of the sizes of prime Selmer groups over a family of cyclic prime twists of non-isotrivial elliptic curves over $\mathbb{F}_q(t)$ satisfying a number of mild constraints conforms to the distribution conjectured by Bhargava, Kane, Lenstra, Poonen, and Rains with explicit error bounds. The key tools used in proving these results are the Riemann hypothesis over global function fields, the Erdös-Kac theorem, and the geometric ergodicity of Markov chains.
