论文标题
在阿贝里安（Abelian
On abelian $2$-ramification torsion modules of quadratic fields
论文作者
论文摘要
对于一个数字字段$ f $和质量$ p $，$ \ mathbb {z} _p $ - torsion模块是Galois组的最大Abelian pro- $ p $ p $ $ f $ f $ p $ p $ p $ f $ to $ f $ the $ f $的扩展，表示为$ \ nathcal {t} _p（f）$ p（f）$ p p $ p p $ p $ p $ p $ p $ p $ p $ p $ p $ p $ p $ p $ p $ p $ p $ p $ p $ p $ p $ p $ p $ p $ p $ p $。在本文中，我们研究了组$ \ Mathcal {t} _2（f）= \ MATHCAL {T} _2（m）$ fimratic Field $ f = \ Mathbb {q}（\ sqrt {m}）$。首先，假设$ M> 0 $，我们证明了一个明确的$ 4 $ -Lank公式，用于$ \ MATHCAL {T} _2（-m）$。此外，应用此公式，我们获得了$ 4 $ -LANK密度的$ \ MATHCAL {T} _2 $ - 假想二次字段的群体。其次，对于$ l $ a奇数，我们获得了$ \ nathcal {t} _2（\ pm l）$和$ \ Mathcal {t} _2（\ pm 2l）$的$ 2 $ - 可见性的结果。特别是我们发现，$ \＃\ MATHCAL {t} _2（l）\ equiv 2 \＃\＃\ Mathcal {t} _2（2l）\ equiv h_2（-2l）\ bmod {16} $ {16} $ IF $ l \ equiv 7 \ equass 7 \ bmod 7 \ bmod {8} $ \ mathbb {q}（\ sqrt {-2L}）$。然后，我们获得$ \ Mathcal {t} _2（\ pm L）$和$ \ Mathcal {t} _2（\ pm 2l）$的密度结果。最后，根据我们的密度结果和数值数据，我们提出分布猜测，当$ f $在任何prime $ p $的真实或想象中的二次范围内变化时，大约是$ \ nathcal {t} _p（f）$，而大约$ \ $ \ natcal {t} _2（\ pm l）$ and $ \ niie $ p pm l）科恩·伦斯特拉（Cohen-Lenstra）启发式方法的精神。我们在$ \ Mathcal {t} _2（l）$ case中的猜想密切连接到Shanks-Sime-Washington关于零2美元的零零$ l $ unctions和基本单位的分布的零分布的猜测。
For a number field $F$ and a prime number $p$, the $\mathbb{Z}_p$-torsion module of the Galois group of the maximal abelian pro-$p$ extension of $F$ unramified outside $p$ over $F$, denoted as $\mathcal{T}_p(F)$, is an important subject in abelian $p$-ramification theory. In this paper we study the group $\mathcal{T}_2(F)=\mathcal{T}_2(m)$ of the quadratic field $F=\mathbb{Q}(\sqrt{ m})$. Firstly, assuming $m>0$, we prove an explicit $4$-rank formula for $\mathcal{T}_2(-m)$. Furthermore, applying this formula, we obtain the $4$-rank density of $\mathcal{T}_2$-groups of imaginary quadratic fields. Secondly, for $l$ an odd prime, we obtain results about the $2$-divisibility of orders of $\mathcal{T}_2(\pm l)$ and $\mathcal{T}_2(\pm 2l)$. In particular we find that $\#\mathcal{T}_2(l)\equiv 2\# \mathcal{T}_2(2l)\equiv h_2(-2l)\bmod{16}$ if $l\equiv 7\bmod{8}$ where $h_2(-2l)$ is the $2$-class number of $\mathbb{Q}(\sqrt{-2l})$. We then obtain density results for $\mathcal{T}_2(\pm l)$ and $\mathcal{T}_2(\pm 2l)$. Finally, based on our density results and numerical data, we propose distribution conjectures about $\mathcal{T}_p(F)$ when $F$ varies over real or imaginary quadratic fields for any prime $p$, and about $\mathcal{T}_2(\pm l)$ and $\mathcal{T}_2(\pm 2 l)$ when $l$ varies, in the spirit of Cohen-Lenstra heuristics. Our conjecture in the $\mathcal{T}_2(l)$ case is closely connected to Shanks-Sime-Washington's speculation on the distributions of the zeros of $2$-adic $L$-functions and to the distributions of the fundamental units.
