Specialization of Forms in the Presence of Characteristic 2: First Steps

We outline a specialization theory of quadratic and (symmetric) bilinear forms with respect to a place λ : K → L∪∞. Here K,L denote fields of any characteristic. We have to make a distinction between bilinear forms and quadratic forms and study them both over fields and valuation rings. For bilinear forms this turns out to be essentially as easy as in the case char L 6= 2, albeit no general cancellation law holds for nondegenerate bilinear forms over a valuation domain O, in which 2 is not a unit. For quadratic forms things are more difficult mainly for two reasons. 1) Forms cannot be diagonalized. 2) The quasilinear part of an anisotropic form over O may become isotropic over the residue class field of O. Nevertheless a somewhat restricted specialization theory for quadratic forms is possible which is good enough to establish a fully fledged generic splitting theory. On the other hand it seems, that for bilinear forms no generic splitting is possible. (Most probably there does not exist a “generic zero field” for a bilinear form over a field of characteristic 2.) But specialization of bilinear forms is nevertheless important for generic splitting of quadratic forms, since a bilinear form and a quadratic form can be multiplied via tensor product to give another quadratic form. All this is explicated in a recent book by the author [Spez]. The book contains more material than outlined here. In particular its last chapter IV gives a specialization theory of forms under “quadratic places”, much more tricky than the theory for ordinary places. Miraculously this leads to a generic splitting theory with respect to quadratic places which is as satisfactory as for ordinary places. If φ is a quadratic form over a field K which has “good reduction” with respect to a place λ : K → L ∪ ∞ then our specialization theory gives a quadratic form λ∗(φ) over L. We also develop a theory of “weak specialization”, which associates to φ only a Witt class λW (φ) of forms over L, but under a more general condition on φ than just having good reduction. In the present article weak specialization plays only an auxiliary role in order to define specializations λ∗(φ). But weak specialization is a key notion in establishing the specialization theory for quadratic places (not described here, cf. [Spez, Chap. IV]).