论文标题
seminarrow class组的2-塞尔默等级的界限
Bounds for 2-Selmer ranks in terms of seminarrow class groups
论文作者
论文摘要
让$ e $为一个数字字段$ k $的椭圆曲线，由一元不可约三立方体多项式$ f（x）$定义。当$ e $在$ k $的所有有限素数上都是\ textit {nice}时，我们将其$ 2 $ -SELMER的排名限制在$ 2 $ -LANK的$ 2 $ -LANK中，该$ 2 $ -LANK是一个修改的理想类$ l = k [x]/{（f（x）} $，我们称之为\ textIt {semi-narrow class $ l $ l $。然后，我们为$ e $提供了一些足够的条件，可以在有限的素数中保持良好状态。 作为一个应用程序，当$ k $是一个真正的二次字段时，$ e/k $是可以半固定的，并且$ f $的判别性是完全负面的，那么我们经常通过计算$ e $的根号$ 2 $ -SELMER等级的$ 2 $ -SELMER等级，以及$ L $的$ 2 $ rank。
Let $E$ be an elliptic curve over a number field $K$ defined by a monic irreducible cubic polynomial $F(x)$. When $E$ is \textit{nice} at all finite primes of $K$, we bound its $2$-Selmer rank in terms of the $2$-rank of a modified ideal class group of the field $L=K[x]/{(F(x))}$, which we call the \textit{semi-narrow class group} of $L$. We then provide several sufficient conditions for $E$ being nice at a finite prime. As an application, when $K$ is a real quadratic field, $E/K$ is semistable and the discriminant of $F$ is totally negative, then we frequently determine the $2$-Selmer rank of $E$ by computing the root number of $E$ and the $2$-rank of the narrow class group of $L$.
