arXiv 2107.030272021-07math.NT

关于由Gross曲线的二次扭曲产生的Selmer群的$λ$ -不变量

On the $λ$-invariant of Selmer groups arising from certain quadratic twists of Gross curves

Jianing Li

备注：: minor revision

摘要：

设$q$与$q \equiv 7 \mod 8$为素数，设$K=\mathbb{Q}(\sqrt{-q})$。然后$2$在$K$中分裂，我们为$2$上面的任何一个素数$K$写$\mathfrak{p}$。令$K_\infty$是在$\mathfrak{p}$外部与第$K_n$层$n$未分支的$K$的唯一$\mathbb{Z}_2$扩展.对于某些二次扩张$F/K$，我们证明了域$F_\infty = FK_\infty$的最大交换2-扩张在$\mathfrak{p}$外无分歧的伽罗瓦群的$λ$ -不变量的一个简单的精确公式.等价地，我们的结果确定了一类高维阿贝尔簇的二次扭转的大族的$F_\infty$上的某些Selmer群的精确$\mathbb{Z}_2$ -余秩，这是标量对定义在$K$的Hilbert类域上的Gross曲线的$K$的限制.我们还讨论了当$λ$ -不变量等于$1$时，$K_n$上相关Selmer群的计算.

英文摘要：

Let $q$ be a prime with $q \equiv 7 \mod 8$, and let $K=\mathbb{Q}(\sqrt{-q})$. Then $2$ splits in $K$, and we write $\mathfrak{p}$ for either of the primes $K$ above $2$. Let $K_\infty$ be the unique $\mathbb{Z}_2$-extension of $K$ unramified outside $\mathfrak{p}$ with $n$-th layer $K_n$. For certain quadratic extensions $F/K$, we prove a simple exact formula for the $λ$-invariant of the Galois group of the maximal abelian 2-extension unramified outside $\mathfrak{p}$ of the field $F_\infty = FK_\infty$. Equivalently, our result determines the exact $\mathbb{Z}_2$-corank of certain Selmer groups over $F_\infty$ of a large family of quadratic twists of the higher dimensional abelian variety with complex multiplication, which is the restriction of scalars to $K$ of the Gross curve with complex multiplication defined over the Hilbert class field of $K$. We also discuss computations of the associated Selmer groups over $K_n$ in the case when the $λ$-invariant is equal to $1$.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

摘要：

英文摘要：

相关论文