构造3-Selmer伙伴族

Selmer companion curves

We say that two elliptic curves E_1, E_2 over a number field K are n-Selmer companions for a positive integer n if for every quadratic character χof K, there is an isomorphism between the n-Selmer groups Sel_n(E_1^χ/K) and Sel_n(E_2^χ/K) of the quadratic twists E_1^χ, E_2^χ. We give sufficient conditions for two elliptic curves to be n-Selmer companions, and give a number of examples of non-isogenous pairs of companions.

2-Selmer near-companion curves

Let $E$ and $A$ be elliptic curves over a number field $K$. Let $χ$ be a quadratic character of $K$. We prove the conjecture posed by Mazur and Rubin on $n$-Selmer near-companion curves in the case $n=2$. Namely, we show if the difference of the $2$-Selmer ranks of $E^χ$ and $A^χ$ is bounded independent of $χ$, there is a $G_K$-isomorphism $E[2] \cong A[2]$.

$p^r$-Selmer companion modular forms

The study of $n$-Selmer group of elliptic curve over number field in recent past has led to the discovery of some deep results in the arithmetic of elliptic curves. Given two elliptic curves $E_1$ and $E_2$ over a number field $K$, Mazur-Rubin\cite{mr} have defined them to be {\it $n$-Selmer companion} if for every quadratic twist $χ$ of $K$, the $n$-Selmer groups of $E_1^χ$ and $E_2^χ$ over $K$ are isomorphic. Given a prime $p$, they have given sufficient conditions for two elliptic curves to be $p^r$-Selmer companion in terms of mod-$p^r$ congruences between the curves. We discuss an analogu…

$p$-扭曲塞尔默近伴随曲线

$p$-twisted Selmer near-companion curves

该研究引入$p$-扭曲塞尔默近伴随（$p$-TSNC）的概念，证明在一定条件下，若数域$K$上的两条椭圆曲线为$p$-TSNC，则对应的分圆域相等，推进了塞尔默近伴随曲线相关猜想的研究。

2-Selmer 伴随模形式

2-Selmer companion modular forms

本文研究了2-Selmer伴随模形式，证明了在剩余Galois表示同构条件下剩余Greenberg 2-Selmer群的同构性，并给出了秩差有界时的逆命题，推广了Mazur-Rubin关于椭圆曲线的结果。

具有任意大2-塞尔默秩的希罗尼亚椭圆曲线的二次扭转

Quadratic Twists of Heronian Elliptic Curves with Arbitrarily Large 2-Selmer Rank

本文确定了面积为2mn的希罗三角形导出的椭圆曲线族的2-塞尔默群，明确其在二次扭转下的变化规律，推广了已有研究，构造出具有任意大2-塞尔默秩的希罗尼亚椭圆曲线显式族。