论文标题
格林伯格猜想的算法复杂性
Algorithmic complexity of Greenberg's conjecture
论文作者
论文摘要
让$ k $是一个完全真实的数字字段，$ p $ a Prime。我们表明，格林伯格的猜想（$λ=μ= 0 $）的``复杂性''是由有限的扭转组$ {\ mathcal t} _k $ $ p $ $ $ p $ pre pre pre pre pre pre pre pre pre pre pre pre pre pre pre pre pre pre pre pre pre pre pre pre $ p pre pre pre pre pre pre pre pre pre pre pre pre pre pre pre pre pre pre pre pre pre pre pre pre pre pre pre pre pre pre pre的$ p $ - $ {\ Mathcal t} _k $ of Ilasers $ k_n $的$ k_n $的$ k_n $（定理（5.2））。这些图像是通过``解开''的正式算法计算获得的。这些图像的等分分配的猜想（5.4）将表明，算法的步骤$ b_n $的限制为$ n \ to \ infty $，因此格林伯格的猜想是iWasawa的唯一框架内的唯一框架，它将具有true````具有prire''``具有可能的''。在$ [k：\ mathbb {q}] $上也没有任何假设，也没有在$ k/\ mathbb {q} $中分解$ p $。
Let $k$ be a totally real number field and $p$ a prime. We show that the ``complexity'' of Greenberg's conjecture ($λ= μ= 0$) is of $p$-adic nature governed (under Leopoldt's conjecture) by the finite torsion group ${\mathcal T}_k$ of the Galois group of the maximal abelian $p$-ramified pro-$p$-extension of $k$, by means of images in ${\mathcal T}_k$ of ideal norms from the layers $k_n$ of the cyclotomic tower (Theorem (5.2)). These images are obtained via the formal algorithm computing, by ``unscrewing'', the $p$-class group of~$k_n$. Conjecture (5.4) of equidistribution of these images would show that the number of steps $b_n$ of the algorithms is bounded as $n \to \infty$, so that Greenberg's conjecture, hopeless within the sole framework of Iwasawa's theory, would hold true ``with probability $1$''. No assumption is made on $[k : \mathbb{Q}]$, nor on the decomposition of $p$ in $k/\mathbb{Q}$.
