arXiv subjects
Publications and source records attributed to Anthony C. Kable.
Let $k$ be a number field and $\tilde k$ a fixed quadratic extension of $k$. In this paper and its companion, we find the mean value of the product of class numbers and regulators of two quadratic extensions $F,F^*\not=\tilde k$ contained in the biquadratic extensions of $k$ containing $\tilde k$.
This is the second part of a two part paper. In this part, we evaluate the previously unevaluated local densities at dyadic places which appear in the density theorem stated in the first part. For this part, we introduce an invariant, the level, attaced to a pair of ramified quadratic extensions of a dyadic local field. This invariant measures how close two fields are in their arithmetic properties and may be of interest independent of its application here.
The purpose of the paper is to prove an analogous correspondence for the above prehomogeneous vector space (1)-(3). For case (1) the correspondence is bijective. However, it turns out that the correspondence is not bijective for cases (2), (3). We describe the fiber structures of this correspondence in subsection 2-4. In section 1 we briefly review basic properties of the non-abelian Galois cohomology set and its relation to the orbit space G_k \ V_k^ss. In subsection 2-4 we consider the prehomogeneous vector spaces (1)-(3) respectively.