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Unimodular lattices over real quadratic fields. (English) Zbl 0807.11021
We investigate integral even unimodular lattices \(L\) in a vector space with a totally positive definite quadratic form, defined over a real quadratic field \(F\). We give explicit constructions of a number of such lattices in dimension 4, for indeterminate field discriminant \(d\) (only depending on \(d\bmod 24\), i.e. on the ramification of 2 and 3 in \(F\)). The lattices we construct here have large automorphism groups. In most cases, the full orthogonal group \(O(L)\) is known (and essentially independent of the field). This is true in particular for the so-called reflective lattices which have a root system of maximal rank. As an application we obtain the full classification (essentially independent of the use of computers) of all even unimodular lattices in dimension 4 over the first 11 real quadratic fields with discriminants \(d=5,8,12,13,17,21,24,28,29,33,37\).

MSC:
11E12 Quadratic forms over global rings and fields
11E41 Class numbers of quadratic and Hermitian forms
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