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 HK of a number field K in totally ramified cyclic p-extensions L/K of degree p^N and Galois group G. Using an elementary property of the algebraic norm in L/K, we show that the kernel of capitulation is in relation with the ``complexity'' of the structure of HL measured via its exponent p^e(L) and the length m(L) of the filtration (HL^i)_i associated to HL as Zp[G]-module. We prove that a sufficient condition of complete capitulation is given by e(L) ∈ [1, N-s] if m(L) ∈ [p^s, p^(s+1)-1] for s ∈ [0, N-1] (Theorem 1.1(i)); this improves the case of ``stability'', #HL = #HK (i.e., m(L)=1, s=0, e(L) = e(K) ≤ N) (Theorem 1.1(ii)). A sufficient condition of partial capitulation is also made explicit. Numerical examples with directly usable PARI programs, showing most often capitulations, are given over cubic fields with p=2 and real quadratic fields with p=3, taking the simplest possible p-extensions L
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https://hal.science/hal-03865383
Soumis le : mardi 22 novembre 2022-11:42:14
Dernière modification le : lundi 11 mars 2024-14:44:05
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Georges Gras. Algebraic norm and capitulation of p-class groups in ramified cyclic p-extensions. 2022. ⟨hal-03865383v1⟩
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