中心单代数上的正交对合与Severi-Brauer簇的函数域
Orthogonal involutions on central simple algebras and function fields of Severi-Brauer varieties
摘要：
中心单代数$A$上的正交对合$σ$，在对$A$的Severi-Brauer簇的函数域$\mathcal{F}(A)$进行标量扩张后，伴随于$\mathcal{F}(A)$上的二次型$q_σ$，该二次型在一个标量因子下唯一定义.对合的一些性质，如双曲性和直到基域的奇数次扩张的各向同性，都被编码在这个二次形式中，这意味着它们对对合$σ$成立当且仅当它们对$q_σ$成立。与此相反，我们证明了存在非完全可分解的正交对合，成为完全可分解的$\mathcal{F}(A)$，使相关的形式$q_σ$是一个Pfister形式。我们还提供了索引$2$代数上的非同构对合的例子，这些对合产生类似的二次型，从而证明了$q_σ$的形式不确定$σ$的同构类，即使底层代数具有索引$2$。因此，我们表明，$e_3$不变的正交对合是不分类的程度$12$，并没有检测到完全可分解的对合程度$16$，而不是发生了什么二次型。
英文摘要：
An orthogonal involution $σ$ on a central simple algebra $A$, after scalar extension to the function field $\mathcal{F}(A)$ of the Severi--Brauer variety of $A$, is adjoint to a quadratic form $q_σ$ over $\mathcal{F}(A)$, which is uniquely defined up to a scalar factor. Some properties of the involution, such as hyperbolicity, and isotropy up to an odd-degree extension of the base field, are encoded in this quadratic form, meaning that they hold for the involution $σ$ if and only if they hold for $q_σ$. As opposed to this, we prove that there exists non-totally decomposable orthogonal involutions that become totally decomposable over $\mathcal{F}(A)$, so that the associated form $q_σ$ is a Pfister form. We also provide examples of nonisomorphic involutions on an index $2$ algebra that yield similar quadratic forms, thus proving that the form $q_σ$ does not determine the isomorphism class of $σ$, even when the underlying algebra has index $2$. As a consequence, we show that the $e_3$ invariant for orthogonal involutions is not classifying in degree $12$, and does not detect totally decomposable involutions in degree $16$, as opposed to what happens for quadratic forms.
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