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.

Meanwhile, in [Y], Yamamura determined all imaginary finite abelian extensions over Q in C with class number 1.

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.

Whilst this problem seems approachable, it is unclear if our techniques can be used for classes of absolutely abelian extensions with unit lattices which lie in `complicated' [A.sub.g,g]-genera such as those referred to in Remark 1.6 (cf.

There are results for abelian extensions of imaginary quadratic fields which are completely analogous to those of Proposition 1.4 and Corollaries 1.10 and 1.12 ([B,H]).

Excluding the field extension types dealt with in Corollaries 1.10 and 1.12 the assertion of (2.4)is also known to be valid for absolutely abelian extensions N/Q of prime degree ([D]), Remark 4.23)

Absolutely abelian extensions of odd prime power conductor.

Holland, On Chinburg's third invariant for abelian extensions of imaginary quadratic fields, Proc.

If e = p then L and K are disjoint, and since L/Q is Galois of degree [p.sup.2], G(KL/Q) is abelian as a composition of 2 abelian extensions. This contradiction forces e = 1, so KL = L and L = L((t)).

Let L/k be an abelian extension that contains K and splits A.

In the situation of Theorem 1, suppose G is the Galois group of an abelian extension of smallest degree splitting the noncrossed product D.