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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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