5
$\begingroup$

Let $Q$ be a quaternion $k$-algebra (namely, a dimension 4 $k$-central simple algebra).

Then it is possible to (canonically) attach a smooth projective conic $C_Q\subseteq \mathbf{P}_k^2$ to $Q$: if $\mathrm{char}\,k \neq 2$, then $Q\simeq \langle x,y : x^2 = a, y^2 = b, xy+yx=0\rangle$ for certain $a\in k,b\in k^\times$ so it suffices to set $C_Q\simeq\{ aX_0^2+bX_1^2 - X_2^2 =0\}\subseteq \mathbb{P}^2_k$. If instead $\mathrm{char}\,k =2$ one has $Q=\langle x,y : x^2+x=a,y^2=b, xy+yx=yy\rangle$ for some $a\in k,b\in k^\times$ and assign $C_Q=\{aX_0^2 +bX_1^2 +X_2^2 + X_0X_2 =0\}$.

There exists an interesting relationship between splitting of quaternion algberas (i.e. $Q\simeq M(2,k)$) and existence of $k$-rational points in the associated conic $C_Q$. This works for every characteristic too.

But there is an even more elegant result, attributed to Witt, which relates isomorphism of two quaternion algebras with the birational equivalence of their associated conics. More precisely:

Theorem. Let $Q,R$ be two quaternion $k$-algebras. Then $Q\simeq R$ if and only if $C_Q$ and $C_R$ are birational (namely $k(C_Q)\simeq k(C_R)$).

Such result can be found, for instance, in the great book by Gille and Szamuely, Central simple algebras and Galois cohomology where, much to my dismay, almost all the results are shown for a field of characteristic $\neq 2$.

In particular, the above Theorem is proved using the definitions for quaternion algebras and conics in characteristic not 2.

I have searched in some more specific book (like Vigneras' lectures and Voight's draft) but this result does not seem to be covered. In the Vigneras' book there is a chapter called "Geometry" in which something similar is done, but the whole chapter works in the non-dyadic case.

I was wondering if such result held for a field of characteristic 2 and, if yes, is there a reference for this?

$\endgroup$
1
  • $\begingroup$ Do you want to have a result for quaternion algebras or for cyclic algebras? The former might not be the 'right' notion if the characteristic is $2$. $\endgroup$ Jun 2, 2018 at 5:19

1 Answer 1

3
$\begingroup$

Norms of quaternion algebras are particular cases of n-fold Pfister forms for n=2 (in any characteristic), and the result holds true for any Pfister forms. See, for example, Elman, Karpenko, Merkurjev "The algebraic and geometric theory of quadratic forms": https://sites.ualberta.ca/~karpenko/publ/Kniga.pdf, Corollary 23.5.

$\endgroup$
0

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.