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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 \(k\).

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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