Wanted: Quadratic Space in Characteristic 2 as a Counterexample to a Theorem of Arf

Question

Hi. Peter Roquette sent me an email asking for an example of a quadratic space in characteristic 2 having certain features. I have no idea on this, but maybe someone reading this does.

He would like an example of a field $K$ of characteristic 2 with the following two properties:

(1) The quaternion algebras over $K$ form a group (within the Brauer group of $K$).

(2) There exists an anisotropic quadratic form of dimension > 4 over $K$ that is "completely regular in the sense of Arf". That means: there is a quadratic space $(V,Q)$ over $K$ such that

(a) $\dim(V) > 4$

(b) if $Q(v) = 0$ then $v = 0$

(c) for the associated bilinear form $B(v,w) = Q(v+w) - Q(v) - Q(w)$, if $B(v,w) = 0$ for all $w$ in $V$ then $v = 0$.

Here's the background. Roquette has written a paper with Falko Lorenz on the historical development of the Arf invariant, and they include in the paper a counterexample to the method of proof of a theorem of Arf. (The paper is at Roquette's homepage at http://www.rzuser.uni-heidelberg.de/~ci3/arf.pdf and it has also appeared in "Mathematische Semesterberichte" vol. 57 (2010) pp. 73--102.) Their counterexample to Arf's method of proof is not actually a counterexample to his main theorem. In order to find a counterexample to the main theorem itself they want a quadratic space with the properties above (in characteristic 2). Roquette has asked around but nobody has yet been able to give him an example.

EDIT (Aug. 16): After Roquette learned about the answer posted below, he has posted on his website http://www.rzuser.uni-heidelberg.de/~ci3/manu.html a paper (number 46) with Lorenz which discusses the solution to his question in the context of Arf's paper. Anyone interested in this topic is encouraged to look at the paper. He wrote to me "Your idea to put my question on the website Math Overflow has worked wonderfully."

Answer 1

I forwarded this question to Detlev Hoffmann, who says that such examples exist. Specifically, you can produce such an example where there is, say, an anisotropic form of dimension 8 using characteristic 2 analogues of the techniques in Merkurjev's 1992 article "Simple algebras and quadratic forms". He says details can be found in the PhD thesis of his student Frederic Faivre at Université Franche-Comté.

Detlev further explained the source of Arf's confusion, which I will now recap. Over a field $F$ of any characteristic, the tensor product of two quaternion algebras is not a division algebra if and only if the two quaternion algebras contain a common quadratic extension. In characteristic different from 2, this is an essentially complete criterion. But in characteristic 2, one has to wonder if this quadratic extension is separable or inseparable.

Draxl showed that if two quaternion algebras contain a common quadratic extension, you can always find one that is separable. That is, the property of containing a common inseparable extension is much stronger (because it implies that they contain a common separable extension). A nice exposition of this can be found in T.Y. Lam's 2002 article "On the linkage of quaternion algebras".

Detlev asserts: If you demand that every pair of quaternion division algebras over $F$ (of characteristic 2) share an inseparable quadratic extension, then there are no anisotropic regular quadratic forms of dimension > 4. "This is essentially due to Baeza. In fact, in some sense even to Arf, except that he didn't realize there's a difference between [sharing separable and inseparable subfields]." So presumably this is what Arf was claiming to have proved.

In contrast to this, the requirement that the quaternion algebras form a subgroup of the Brauer group is just that every pair of quaternion division algebras share a separable quadratic extension. This is a much weaker hypothesis, and allows for examples of fields like in Faivre's thesis.

Here are some precise references for the theorem "every pair of quaternion algebras over $F$ share an inseparable quadratic extension implies every regular quadratic form of dimension > 4 is isotropic":

• Arf (with confusion mentioned above): Satz 11 in "Untersuchungen über quadratische Formen in Körpern der Charakteristik 2. I." J. reine angew. Math. 183, 148-167 (1941)"
• Baeza: Theorem 3.1 in "Comparing $u$-invariants of fields of characteristic $2$." Bol. Soc. Brasil. Mat. 13 (1982), no. 1, 105--114.
• Faivre's thesis: Proposition 3.3.5 (with complete proof)