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Realization of symmetric \(\mathbb Z\)-bilinear forms as scaled Hermitian trace forms. (Réalisation de formes \(\mathbb Z\)-bilinéaires symétriques comme formes trace hermitiennes amplifiées.) (French) Zbl 1014.11030

Given a nondegenerate symmetric \(\mathbb Z\)-bilinear form of even rank, not \(\mathbb Q\)-isomorphic to the hyperbolic plane, the author shows that there is an algebraic integer \(\alpha\) such that the form can be seen as a scaled Hermitian trace form (Tr\(_{\mathbb Q(\alpha)/\mathbb Q}(\lambda v_i v_j^\sigma)\)) of the algebra \(\mathbb Z[\alpha]\). The proof is by explicit construction. This is inspired by a result of M. Krüskemper [Algebraic construction of bilinear forms over \(\mathbb Z\), Publ. Math. Besançon, Théorie des nombres, Années 1996/97-1997/98, 4 p. (1999)] who showed that every nondegenerate symmetric \(\mathbb Z\)-bilinear form can be realized as a scaled trace form for some algebra \(\mathbb Z[\alpha]\). The author gives an explicit method to find \(\alpha\), the involution \(\sigma\) on \(\mathbb Z[\alpha]\), the appropriate bases \(\{v_i\}, \{v_i'\}\) and the scaling factor \(\lambda\). He illustrates his method by applying it to the lattice \(\mathbb A_4\).

MSC:

11E39 Bilinear and Hermitian forms
11E12 Quadratic forms over global rings and fields
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References:

[1] Bayer-Fluckiger, E., Jacques Martinet, Formes quadratiques liées aux algèbres semi-simples. J. reine angew. Math.451 (1994), 51-69. · Zbl 0801.11020
[2] Krüskemper, M., Algebraic construction of bilinear forms over Z. Pub. Math. de Besançon, Théorie des nombres (96/97-97/98). · Zbl 0689.10027
[3] Taussky, O., On a theorem of Latimer and MacDuffee. Canad. J. Math.1 (1949), 300-302. · Zbl 0045.15404
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