论文标题
$ 3 $ -Selmer Group，理想的课程组和立方体总和问题
$3$-Selmer group, ideal class groups and cube sum problem
论文作者
论文摘要
考虑一个mordell曲线$ e_a：y^2 = x^3+a $，$ a \ in \ mathbb z $。这些曲线具有合理的$ 3 $发育性，例如$φ$。我们在$ \ mathbb q（ζ_3）$ $ 3 $ - $ 3 $ - $ 3 $的$ 3 $ - $ \ MATHBB Q（ζ_3）$的$ 3 $ - 部分的$ 3 $ - $ 3 $零件的$ e_a $的$ e_a $组中的上限和下限。使用我们在Selmer组上的界限，我们证明了有理立方体总和问题的某些情况。此外，使用这些边界，我们给出了Mordell曲线的明确家庭，以表明$ e_a $，$ {\ rm sel}^3（e_ {a}/\ mathbb q）= 0 $（分别$ {\ rm sel}^3（e_}^3）
Consider a Mordell curve $E_a:y^2=x^3+a$ with $a \in \mathbb Z$. These curves have a rational $3$-isogeny, say $φ$. We give an upper and a lower bound on the rank of the $φ$-Selmer group of $E_a$ over $\mathbb Q(ζ_3)$ in terms of the $3$-part of the ideal class group of certain quadratic extension of $\mathbb Q(ζ_3)$. Using our bounds on the Selmer groups, we prove some cases of the rational cube sum problem. Further, using these bounds, we give explicit families of the Mordell curves to show that for a positive proportion of $E_a$, ${\rm Sel}^3(E_{a}/\mathbb Q)=0$ (respectively ${\rm Sel}^3(E_{a}/\mathbb Q)$ has $\mathbb F_3$-rank $1$).
