论文标题
2-Selmer Group助理的等级为$θ$ -CONGRUENT数字
The rank of 2-Selmer group associate to $θ$-congruent numbers
论文作者
论文摘要
我们研究了与$π/3 $和$2π/3 $ - 千万的数字相关的$ 2 $ - $ {\ rm selmer} $组的均等。我们的第二个结果给出了大约$π/3 $和$2π/3 $非统一数字的一些正密度，这些数字可以支持戈德菲尔德的猜想的偶数。我们提供了一些必要的条件，以便$ n $是椭圆曲线$ e_n $的$π/3 $ - 综合号码，其shafarevich-tate群体是非平凡的。在上一节中，我们表明，对于$ n = pq \ equiv 5（resp。\ 11）\ pmod {24} $，非$π/3 $的密度（$resp。$ $ $2π/3 $） - 一致数量至少为75 \％，在$ p，$ p，q $的情况下。
We study the parity of rank of $2$-${\rm Selmer}$ groups associated to $π/3$ and $2π/3$-congruent numbers. Our second result gives some positive densities about $π/3$ and $2π/3$ non-congruent numbers which can support the even part of Goldfeld's conjecture. We give some necessary conditions such that $n$ is non $π/3$-congruent number for elliptic curves $E_n$ whose Shafarevich-Tate group is non-trivial. In the last section, we show that for $n=pq\equiv 5(resp. \ 11)\pmod{24}$, the density of non $π/3$($resp.$ $2π/3$)-congruent numbers is at least 75\%, where $p,q$ are primes.
