Spinor genera under field extensions. IV: Spinor class fields.

*(English)*Zbl 0725.11019This paper appears as the fourth in a series dealing with the behavior of spinor genera of quadratic forms under extensions of the underlying field of scalars (the first three papers in the series [Acta Arith. 32, 115-128 (1977; Zbl 0393.10020); Am. J. Math. 100, 523-538 (1978; Zbl 0393.10021); Number theory and algebra, 43-62 (1977; Zbl 0393.10022)] were written by the second author and the reviewer).

But the title of this paper is perhaps not fully indicative of its scope. In it, the authors introduce an interesting new concept, the “spinor class field” associated to a genus of quadratic forms, which is an extension field of the original scalar field defined via class field theory. More specifically, let L be a quadratic R-lattice of rank \(n\geq 3\), where R is the ring of integers of a number field F, and let G be the genus of L. The proper spinor genera in G are the orbits of the lattices in G under the action of a certain normal subgroup of the split rotation group \(J_ V\) of the underlying quadratic space V. Via a mapping induced by the spinor norm function, the corresponding quotient group of \(J_ V\) is isomorphic to \(J_ F/P_ DJ^ L_ F\), where \(J_ F\) is the idèle group of F, \(P_ D\) is the subgroup of principal idèles generated by elements which are positive with respect to all real spots on F at which the localization of V is anisotropic, and \(J^ L_ F\) is determined by the images of the local spinor norm mappings. The subgroup \(P_ DJ^ L_ F\) does not in general contain the subgroup \(P_ F\) consisting of all principal idèles. The key step in the authors’ definition of the spinor class field is the observation that there exists a naturally associated open subgroup H(L) of \(J_ F\) which does contain \(P_ F\) and has the property that \(J_ F/P_ DJ^ L_ F\) is isomorphic to \(J_ F/H(L)\). Then, by class field theory, there exists a unique abelian extension \(\Sigma\) of F such that H(L) equals \(P_ FN_{\Sigma /F}(J_{\Sigma})\). This field \(\Sigma\) is the spinor class field associated to the genus of L.

By the Artin Reciprocity Law, the Galois group Gal(\(\Sigma\) /F) is canonically isomorphic to \(J_ F/H(L)\), which is known from the spinor genus theory to be an elementary abelian 2-group. Hence, it follows from Kummer theory that \(\Sigma\) is a multiquadratic extension of F. Conversely, it is proven that every multiquadratic field of some genus of lattices defined over F.

The link to the earlier papers in the series comes through the application of the concept of the spinor class field to the study of the behavior of spinor genera under lifting. Let E be a finite extension field of F and denote by \(\tilde L\) the lifted lattice \(L\otimes_ RS\), where S is the ring of integers of E. Under the assumption that 2 is unramified in F, it is proven that the number of spinor genera in the genus of \(\tilde L\) is at least as large as the number of spinor genera in the genus of L, whenever E and the spinor class field associated to L are linearly disjoint over F. As a special case, this yields Theorem 3.5 in the third paper of the series cited above. In the final section, an analogue of the Principal Ideal Theorem is proved. For a genus G of lattices defined over F, an extension field of F is constructed with the property that all lattices in G become spinor-equivalent when lifted to this field.

The spinor class field provides a new perspective in which to view spinor generic aspects of quadratic form theory. It promises to be quite interesting to see what new insights this point of view reveals, and to what extent the powerful machinery of class field theory can be brought to bear via this approach on problems in integral quadratic form theory.

But the title of this paper is perhaps not fully indicative of its scope. In it, the authors introduce an interesting new concept, the “spinor class field” associated to a genus of quadratic forms, which is an extension field of the original scalar field defined via class field theory. More specifically, let L be a quadratic R-lattice of rank \(n\geq 3\), where R is the ring of integers of a number field F, and let G be the genus of L. The proper spinor genera in G are the orbits of the lattices in G under the action of a certain normal subgroup of the split rotation group \(J_ V\) of the underlying quadratic space V. Via a mapping induced by the spinor norm function, the corresponding quotient group of \(J_ V\) is isomorphic to \(J_ F/P_ DJ^ L_ F\), where \(J_ F\) is the idèle group of F, \(P_ D\) is the subgroup of principal idèles generated by elements which are positive with respect to all real spots on F at which the localization of V is anisotropic, and \(J^ L_ F\) is determined by the images of the local spinor norm mappings. The subgroup \(P_ DJ^ L_ F\) does not in general contain the subgroup \(P_ F\) consisting of all principal idèles. The key step in the authors’ definition of the spinor class field is the observation that there exists a naturally associated open subgroup H(L) of \(J_ F\) which does contain \(P_ F\) and has the property that \(J_ F/P_ DJ^ L_ F\) is isomorphic to \(J_ F/H(L)\). Then, by class field theory, there exists a unique abelian extension \(\Sigma\) of F such that H(L) equals \(P_ FN_{\Sigma /F}(J_{\Sigma})\). This field \(\Sigma\) is the spinor class field associated to the genus of L.

By the Artin Reciprocity Law, the Galois group Gal(\(\Sigma\) /F) is canonically isomorphic to \(J_ F/H(L)\), which is known from the spinor genus theory to be an elementary abelian 2-group. Hence, it follows from Kummer theory that \(\Sigma\) is a multiquadratic extension of F. Conversely, it is proven that every multiquadratic field of some genus of lattices defined over F.

The link to the earlier papers in the series comes through the application of the concept of the spinor class field to the study of the behavior of spinor genera under lifting. Let E be a finite extension field of F and denote by \(\tilde L\) the lifted lattice \(L\otimes_ RS\), where S is the ring of integers of E. Under the assumption that 2 is unramified in F, it is proven that the number of spinor genera in the genus of \(\tilde L\) is at least as large as the number of spinor genera in the genus of L, whenever E and the spinor class field associated to L are linearly disjoint over F. As a special case, this yields Theorem 3.5 in the third paper of the series cited above. In the final section, an analogue of the Principal Ideal Theorem is proved. For a genus G of lattices defined over F, an extension field of F is constructed with the property that all lattices in G become spinor-equivalent when lifted to this field.

The spinor class field provides a new perspective in which to view spinor generic aspects of quadratic form theory. It promises to be quite interesting to see what new insights this point of view reveals, and to what extent the powerful machinery of class field theory can be brought to bear via this approach on problems in integral quadratic form theory.

Reviewer: A.G.Earnest (Carbondale)