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Uniform distribution of the Steinitz invariants of quadratic and cubic extensions. (English) Zbl 1113.11065
Let $$k$$ be a number field with ring of integers $$\mathcal O_k$$ and class group $$\text{Cl}(k)$$ of order $$h$$. Let $$K/k$$ be a finite extension; one can associate to the ring of integers $$\mathcal O_K$$ of $$K$$ an element $$S(\mathcal O_K)$$ of $$\text{Cl}(k)$$, i.e, the Steinitz invariant of $$\mathcal O_K$$ as $$\mathcal O_k$$-module. The aim of the paper is to show an equidistribution result for the values of this map when restricted to all extensions $$K/k$$ of a fixed degree $$d$$, ordered according to the size of the absolute ideal norm $$\mathcal N(\Delta_{K/k})$$ of their relative discriminant $$\Delta_{K/k}$$.
More precisely, the authors prove the following
Theorem: For $$d=2,3$$ and for any $$\mathcal C\in \text{Cl}(k)$$ one has $\lim_{B\to\infty} {\left|\{K| [K:k]=d,\, \mathcal N(\Delta_{K/k})\leq B,\, S(\mathcal O_K) = \mathcal C\}\right| \over \left| \{\{K| [K:k]=d,\, \mathcal N(\Delta_{K/k})\leq B\}\right| } = {1\over h}.$ For the case $$d=2$$ this is a refinement of a previous result of A. Fröhlich [Mathematika 7, 15–22 (1960; Zbl 0229.12007)], while for $$d=3$$ the result is completely new. The main ingredients of the proof for $$d=3$$ are the Shintani zeta function of the space of binary cubic forms and the Delone-Faddeev correspondence between a suitable subset of the binary cubic forms defined over $$k$$ and the cubic extensions of $$k$$.

##### MSC:
 11R45 Density theorems 11R11 Quadratic extensions 11R16 Cubic and quartic extensions 11S90 Prehomogeneous vector spaces 11M41 Other Dirichlet series and zeta functions
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