Quadratic forms related to semisimple algebras. (Formes quadratiques liées aux algèbres semi-simples.)(French)Zbl 0801.11020

This paper deals mainly with lattices which arise from fractional ideals of maximal orders in a simple $$\mathbb{Q}$$-algebra $$A$$ endowed with the symmetric bilinear form $$(x,y)\mapsto \text{Trd}(\alpha x\overline{y})$$, when $$A$$ is a totally real number field or a CM-field or a totally definite quaternion algebra, $$\alpha$$ is a totally positive element in the maximal real subfield and $$x\to \overline{x}$$ is the standard involution (the identity in the first case), so that the corresponding quadratic form is positive definite.
We find in §2 the possible discriminants for such lattices and we study in §3 their parity. Section 4 is devoted to the construction of known lattices, which appear here together with particular algebraic structures, including the Coxeter-Todd lattice $$K_{12}$$, the Leech lattice $$\Lambda_{24}$$, the root lattice $$E_ 6\perp E_ 6$$ over some quaternion algebra, …The paper ends with an appendix devoted to the cyclotomic properties of root lattices.
Section 1 is special. We make use of quadratic forms which are not assumed to be positive, and we use the known property of the signature modulo 8 of integral quadratic forms to characterize the number fields whose different is in the square of a narrow class (Théorème 1.8). (This theorem is also a consequence of a theorem of J. V. Armitage [Invent. Math. 2, 238-246 (1967; Zbl 0143.063)]).

MSC:

 11E12 Quadratic forms over global rings and fields 11R52 Quaternion and other division algebras: arithmetic, zeta functions

Zbl 0143.063
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