论文标题
Chevalley-Herbrand配方和真正的Abelian主要猜想
The Chevalley-Herbrand formula and the real abelian Main Conjecture
论文作者
论文摘要
Abelian领域的主要定理（尽管在大多数情况下证明了证据，但通常称为主要猜想）具有悠久的历史，它通过“基本算术”找到了解决方案，正如华盛顿的书中详细介绍了Thaine的方法，导致了Kolyvagin的Euler Systems。真正的Abelian字段的分析理论$ k $说（在半简单的情况下），$ p $ c $ class group $ \ mathcal {h} _k $等于$ p $ intex cyclotomic units $（\ Mathcal {\ Mathcal {e}我们已经猜想（1977）关系$ \＃\ MATHCAL {h}_φ=（\ Mathcal {e}_φ：\ Mathcal {f}_φ）$用于使用nordredicable $ p $ p $ - addic $ p $ - adic $ - addic $ p $ - ad的$ - ad $ p $ - ad的$ p $ - addic组件的$。我们在本文中开发了以下新的有希望的链接：（i）在$ p $ - extensions $ l/k $，$ l \ subset k（μ__\ ell^{}）中给出``模棱两可的'''''' （ii）$ \ Mathcal {h} _k $ in $ l $的投降现象; （iii）真正的主要猜想$ \＃\ m nathcal {h}_φ=（\ Mathcal {e}_φ：\ Mathcal {f}_φ）$ for All〜 $φ$。我们证明，只要$ \ Mathcal {h} _k $ capitulates在$ l $中（theorem \ ref ref {thmppl}），真正的主要猜想就会实现。使用Pari计划的计算支持主要猜想的新理念。投降的非常频繁的现象表明猜想1.2。
The Main Theorem for abelian fields (often called Main Conjecture despite proofs in most cases) has a long history which has found a solution by means of "elementary arithmetic", as detailed in Washington's book from Thaine's method having led to Kolyvagin's Euler systems. Analytic theory of real abelian fields $K$ says (in the semi-simple case) that the order of the $p$-class group $\mathcal{H}_K$ is equal to the $p$-index of cyclotomic units $(\mathcal{E}_K : \mathcal{F}_K)$. We have conjectured (1977) the relations $\# \mathcal{H}_φ= (\mathcal{E}_φ: \mathcal{F}_φ)$ for the isotypic $p$-adic components using the irreducible $p$-adic characters $φ$ of $K$. We develop, in this article, new promising links between: (i) the Chevalley-Herbrand formula giving the number of ``ambiguous classes'' in $p$-extensions $L/K$, $L \subset K(μ_\ell^{})$ for the auxiliary prime numbers $\ell \equiv 1 \pmod {2p^N}$ inert in $K$; (ii) the phenomenon of capitulation of $\mathcal{H}_K$ in $L$; (iii) the real Main Conjecture $\# \mathcal{H}_φ= (\mathcal{E}_φ: \mathcal{F}_φ)$ for all~$φ$. We prove that the real Main Conjecture is trivially fulfilled as soon as $\mathcal{H}_K$ capitulates in $L$ (Theorem \ref{thmppl}). Computations with PARI programs support this new philosophy of the Main Conjecture. The very frequent phenomenon of capitulation suggests Conjecture 1.2.
