Algebraic norm and capitulation of p-class groups in ramified cyclic p-extensions
Résumé
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 ≥ 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) ∈ [1, N-s(L)] if m(L) ∈ [p^s(L), p^(s(L)+1)-1] for s(L) ∈ [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(µ_ℓ), with primes ℓ=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 ℓ's are proposed (Conjectures 1.4, 2.4). Capitulation property of other arithmetic invariants is examined.
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