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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:
 1.1e+13 Quadratic forms over global rings and fields 1.1e+42 Class numbers of quadratic and Hermitian forms
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##### References:
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