On the p-adic Leopoldt transform of a power series

On the p-adic Leopoldt transform of a power series BrunoAngì es 1 Let p be an odd prime number. Let X be the projective limit for the norm maps of the p-Sylow subgroups of the ideal class groups of Q(ζ p n+1), n ≥ 0. Let ∆ = Gal(Q(ζ p)/Q) and let θ be an even and non-trivial character of ∆. Then X is a Z p [[T ]]-module and the characteristic ideal of the isotypic component X(ωθ −1) is generated by a power series f (T, θ) ∈ Z p [[T ]] such that (see for example [2]): ∀n ≥ 1, n ≡ 0 (mod p − 1), f ((1 + p) 1−n − 1, θ) = L(1 − n, θ), where L(s, θ) is the usual Dirichlet L-series. Therefore, it is natural and interesting to study the properties of the power series f (T, θ). We denote by f (T, θ) ∈ F p [[T ]] the reduction of f (T, θ) modulo p. Then B. Ferrero and L. Washington have proved ([3]): f (T, θ) = 0. Note that, in fact, we have ([1]): f (T, θ) ∈ F p [[T p ]]. W. Sinnott has proved the following ([8]): f (T, θ) ∈ F p (T). But, note that ∀a ∈ Z * p , F p [[T ]] = F p [[(1 + T) a − 1]]. Therefore it is natural to introduce the notion of a pseudo-polynomial which is an element F (T) in F p [[T ]] such that there exist an integer r ≥ 1, c 1 , · · · c r ∈ F p , a 1 , · · · , a r ∈ Z p , such that F (T) = r i=1 c i (1 + T) a i. An element of F p [[T ]] will be called a pseudo-rational function if it is the quotient of two pseudo-polynomials. In this paper, we prove that f (T, θ) is not a pseudo-rational function (part 1) of Theorem 4.5). This latter result suggests the following question: is f (T, θ) algebraic over F p (T)? We suspect that this is not the case but we