论文标题
在格林伯格的广泛猜想上
On Greenberg's generalized conjecture
论文作者
论文摘要
对于数字字段$ f $和一个奇数$ p，$ $ \ tilde {f} $是所有$ \ mathbb {z} _p $ - extensions $ f $ and $ \tildeλ$的合成。令$ g_ {s}（\ tilde {f}）$是最大扩展的$ \ tilde {f} $的Galois组，该{f} $是在$ p $ - addic和无限位置之外未经夸大的。在本文中，我们研究$ \tildeλ$ -module $ \ mathfrak {x} _ {s}}^{（ - i）}（\ tilde {f}）：= h_1（g_s（g_s（\ tilde {f}）， $ x（\ tilde {f}（μ_p））（i-1）^δ，$δ：= \ m m缩 abelian unramified pro-$p$-extension of $\tilde{F}(μ_{p}).$ More precisely, we show that under a decomposition condition, the pseudo-nullity of the $\tildeΛ$-module $X(\tilde{F}(μ_p))(i-1)^Δ$ is implied by the existence of a $ \ mathbb {z} _ {p}^d $ - extension $ l $带有$ \ mathfrak {x} _ {s}^{（ - i）}（ - i）}（l）：= h_1（g_s（l），\ mathbb {z} $ \ mathbb {z} _ {p} $ - 扩展$ f _ {\ infty} $满足$ h^2（g_ {s}（f _ {\ infty}），\ Mathbb {q} _ {q} _ {p}/\ mathbb {z}格林伯格的广义猜想当整数$ i \ equiv 1 \ mod {[f（μ_p）：f]}。$对于$（p，i）$ - 常规字段实现。
For a number field $F$ and an odd prime number $p,$ let $\tilde{F}$ be the compositum of all $\mathbb{Z}_p$-extensions of $F$ and $\tildeΛ$ the associated Iwasawa algebra. Let $G_{S}(\tilde{F})$ be the Galois group over $\tilde{F}$ of the maximal extension which is unramified outside $p$-adic and infinite places. In this paper we study the $\tildeΛ$-module $\mathfrak{X}_{S}^{(-i)}(\tilde{F}):=H_1(G_S(\tilde{F}), \mathbb{Z}_p(-i))$ and its relationship with $X(\tilde{F}(μ_p))(i-1)^Δ,$ the $Δ:=\mathrm{Gal}(\tilde{F}(μ_{p})/\tilde{F})$-invariant of the Galois group over $\tilde{F}(μ_{p})$ of the maximal abelian unramified pro-$p$-extension of $\tilde{F}(μ_{p}).$ More precisely, we show that under a decomposition condition, the pseudo-nullity of the $\tildeΛ$-module $X(\tilde{F}(μ_p))(i-1)^Δ$ is implied by the existence of a $\mathbb{Z}_{p}^d$-extension $L$ with $\mathfrak{X}_{S}^{(-i)}(L):=H_1(G_S(L), \mathbb{Z}_p(-i))$ being without torsion over the Iwasawa algebra associated to $L,$ and which contains a $\mathbb{Z}_{p}$-extension $F_{\infty}$ satisfying $H^2(G_{S}(F_{\infty}),\mathbb{Q}_{p}/\mathbb{Z}_p(i))=0.$ As a consequence we obtain a sufficient condition for the validity of Greenberg's generalized conjecture when the integer $i\equiv 1 \mod{[F(μ_p):F]}.$ This existence is fulfilled for $(p, i)$-regular fields.
