Substitutions de séries formelles et cyclotomie Original Research Article

Let imageqbe the finite field withq=pfelements. LetΓbe the closed subgroup generated by a seriesγ(t)set membership, variantt+t2imageq[[t]] of infinite order for the substitution law, and letΛ=imagep[[Γ]] be the Iwasawa algebra ofΓ. The multiplicative groupU=1+timageq[[t]] is a rightΓ-module by substitution. We give a necessary and sufficient condition on the ramification ofγ(t) for which theΛ-moduleUis noetherian and, in this case, we describe its structure. In particular, we prove thatUpossesses a non trivialΛ-torsion if and only if there exists a seriess(t)set membership, varianttimageq[[t]] and ap-adic unit u such thatsring operatorγ(t)=(1+s(t))u−1.