论文标题
p-rank $ε$ - 集体上的注射件对于p-extensions的塔是正确的
The p-rank $ε$-conjecture on class groups is true for towers of p-extensions
论文作者
论文摘要
令P $ \ ge $ 2为给定的质量数。我们证明，对于任何数字字段kappa和任何整数e $ \ ge $ 1，p级$ε$ - 注射器，在p级cl \ _f上，为家庭f \ _kappa^p^e hird tot f/kappa构建为连续p/kappa的连续程度p cyclic扩展（没有其他条件）（例如，没有其他条件）。 ＃（cl \ _f [p]）<< \ _ {kappa，p^e，$ε$}（$ \ sqrt $ d \ _f）^$ε$，其中d \ _f是歧视剂的绝对值（theorem 3.6）和更普遍的，更普遍的，更普遍的，更普遍的， ＃（cl \ _f [p^r]）<< \ _ {kappa，p^e，$ε$}（$ \ sqrt $ d \ _f）^$ε$，对于任何r $ \ ge $ 1固定。本说明概括了家庭f \ _q^p的情况（属理论和$ε$ - 注射p级群体，J。 Then we prove, for F\_kappa^p^e, the p-rank $ε$-conjecture on the cohomology groups H^2(G\_F,Z\_p) of Galois p-ramification theory over F (Theorem 4.3) and for some other classical finite p-invariants of F, as the Hilbert kernels and the logarithmic class groups.
Let p$\ge$2 be a given prime number. We prove, for any number field kappa and any integer e$\ge$1, the p-rank $ε$-conjecture, on the p-class groups Cl\_F, for the family F\_kappa^p^e of towers F/kappa built as successive degree p cyclic extensions (without any other Galois conditions) such that F/kappa be of degree p^e, namely: #(Cl\_F[p])<<\_{kappa,p^e,$ε$}($\sqrt$D\_F)^$ε$, where D\_F is the absolute value of the discriminant (Theorem 3.6) and, more generally, #(Cl\_F[p^r])<<\_{kappa,p^e,$ε$}($\sqrt$D\_F)^$ε$, for any r$\ge$1 fixed. This Note generalizes the case of the family F\_Q^p (Genus theory and $ε$-conjectures on p-class groups, J. Number Theory 207, 423--459 (2020)), whose techniques appear to be "universal" for all relative degree p cyclic extensions and use the Montgomery--Vaughan result on prime numbers. Then we prove, for F\_kappa^p^e, the p-rank $ε$-conjecture on the cohomology groups H^2(G\_F,Z\_p) of Galois p-ramification theory over F (Theorem 4.3) and for some other classical finite p-invariants of F, as the Hilbert kernels and the logarithmic class groups.
