Let N/K be a finite abelian extension of number fields of group G.

Let N/K be an abelian extension of number fields of group G, and let H be a subgroup of G.

We denote by [Z.sub.l] the ring of l-adic integers, and by [B.sub.[infinity]] the unique abelian extension of Q in C whose Galois group over Q is topologically isomorphic to the additive group of Z/.

Next let F be a finite abelian extension over Q in C.

Let L/k be an

abelian extension that contains K and splits A.

But [square root of [DELTA]] [member of] [Q.sup.ab] = [Q.sup.cyc], where [Q.sup.ab] is the maximal

abelian extension of Q.

Then by Proposition 1.6, [L.sub.0][K.sub.[infinity]] is an

abelian extension of M.

For example, one may cite the study of the maximal unramified extension of a local field or the maximal

abelian extension of a global field.

Finally in the last section we reach a factorization formula which is an analog of the decomposition of the Dedekind zeta function of an

abelian extension into Hecke L-functions.

Kato ([5,6] and [7]) the Galois group of an

abelian extension field on a q-dimensional local field K is described by the Milnor K-group [K.sup.M.sub.q] (K) for q [greater than or equal to] [1.sup.*1] The information on the ramification is related to the natural filtration [U.sup.m] [K.sub.q] = [U.sup.m] [K.sup.M.sub.q] (K) which is by definition the subgroup generated by {1 + [m.sup.m.sub.K],[K.sup.x], ..., [K.sup.x]},where [m.sub.K] is the maximal ideal of the ring of integers [O.sub.k].

Among the topics are quadratic points of classical modular curves, p-adic point counting on singular super-elliptic curves, a vanishing criterion for Dirichlet series with periodic coefficients, the Sato-Tate conjecture for a Picard curve with a complex multiplication, arithmetic twists with

abelian extensions, and transcendental numbers with special values of Dirichlet series.