On some modules attached to the Lubin-Tate formal groups.
Let \(p\) be a prime number, \({\mathbb C}_p\) the completion of the algebraic closure of \({\mathbb Q}_p\) with the ring of integers \({\mathcal O}\). Let \(F(X,Y)\) be a Lubin-Tate formal group over \({\mathbb Z}_p\) and \(h(X)\) a meromorphic series in \({\mathcal O}((X))^\times\). Let \(\chi\) be a primitive Dirichlet character with conductor a power of \(p\). In an earlier work [Kyushu J. Math. 51, No. 2, 381--391 (1997; Zbl 0909.11053)], the first author constructed a meromorphic function \(L_p(s,\chi,F,h)\) in the case that \(F(X,Y)\) is a relative Lubin-Tate formal group defined over the ring of integers of an unramified extension of \({\mathbb Q}_p\). If \(F(X,Y)\) is the formal multiplicative group \(G_m(X,Y)=(X+1)(Y+1)-1\) and \(h(X)=X\), then \(L_p(s,\chi,G_m,X)\) coincides with the Kubota-Leopoldt \(p\)-adic \(L\)-function \(L_p(s,\chi)\). Iwasawa proved that \(L_p(s,\chi)\) is related to the Galois structure of the local units modulo the closure of the cyclotomic units. The main purpose of this paper is to generalize the result of Iwasawa for the function \(L_p(s,\chi,F,h)\), in the case that \(F\) is defined over \( {\mathbb Z}_p\) and that \(h(X)\) satisfies certain appropriate conditions.
scientific article; zbMATH DE number 1744150
scientific article