Algebraic norm and capitulation of p-class groups in ramified cyclic p-extensions

Abstract

We examine the phenomenon of capitulation of the $p$-class group $H_K$ of a real number field $K$ in totally ramified cyclic p-extensions $L/K$ of degree $p^N$. Using an elementary property of the algebraic norm $\nu_{L/K}$, we show that the kernel of capitulation is in relation with the "complexity" of the structure of $H_L$ measured via its exponent $p^e(L)$ and the length $m(L)$ of the usual filtration $\{H_L^i\}_{i \ge 0}$ associated to $H_L$ as $Z_p[Gal(L/K)]$-module. We prove that a sufficient condition of capitulation is given by $e(L) \in [1, N-s(L)]$ if $m(L) \in [p^s(L), p^(s(L)+1)-1]$ for $s(L) \in [0, N-1]$ (Theorem 1.1); this improves the case of "stability" $\#H_L = \#H_K$ (i.e., $m(L) = 1$, $s(L)=0$, $e(L) = e(K)$) (Theorem 1.2). Numerical examples (with PARI programs) showing most often capitulation of $H_K$ in $L$, are given, taking the simplest abelian $p$-extensions $L < K(\mu_\ell$), with primes $\ell=1$ (mod $2p^N$) over cubic fields with $p=2$ and real quadratic fields with $p=3$. Some conjectures on the existence of non-zero densities of such $\ell$'s are proposed (Conjectures 1.4, 2.4). Capitulation property of other arithmetic invariants is examined.