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.
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.
For example, we show that the `fine structure' results of [F4] are for the special classes of abelian extensions considered there the best approximations to a proof of Chinburg's conjecture which can be obtained without analysis of extension class data.
Our main aim in this paper, however, is to show that for certain special classes of prime power degree abelian extensions one can obtain much finer information by working modulo Swan subgroups [T.
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.
There are results for abelian extensions of imaginary quadratic fields which are completely analogous to those of Proposition 1.
Therefore to find bounds for the degree of a smallest abelian splitting field of D, it suffices to consider finite abelian extensions of k that contain K and split A.
2], G(KL/Q) is abelian as a composition of 2 abelian extensions.
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.