发表机构
Yokohama National University（横滨国立大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文给出 Hermitian 簇上 Hodge 型标准猜想的显式初等证明，利用对偶极图控制相交形式，并证明 $p$-adic 初等因子整除 $q^m$，从而得到代数环的数值等价性及猜想对有限覆盖簇的推广。
AI 中文摘要
设 $X$ 是有限域上的偶维数非奇异 Hermitian 簇，设 $\Lambda$ 是由其定义在基域上的极大全迷向线性子空间类张成的格。根据 Dummigan、Dummigan--Tiep 和 Shimada 的工作，这些类张成中维上同调，且原始部分上的相交形式是正定的，正如 Hodge 型标准猜想所预言的那样。我们给出一个初等证明，该证明避免了有限酉群的表示论，仅使用与 Hermitian 形式相伴的对偶极图。它表明原始部分上的相交形式是欧几里得内积的显式正倍数。新证明的优势在于它还能在 $p$-adic 意义上控制 $\Lambda$。若 $X$ 定义在 $\F_{q^2}$ 上且维数为 $2m$，则这些子空间的相交矩阵的每个 $p$-adic 初等因子都整除 $q^m$。因此，$q^m$ 乘以每个余维数为 $m$ 的代数环在数值上等价于它们的整系数线性组合。作为副产品，Hodge 型标准猜想对有限覆盖为 Hermitian 簇的簇成立，且 $p$-adic 界沿次数与 $p$ 互素的有限态射下降。
英文摘要
Let $X$ be a nonsingular Hermitian variety of even dimension over a finite field, and let $Λ$ be the lattice spanned by the classes of its maximal totally isotropic linear subspaces defined over the base field. By work of Dummigan, Dummigan--Tiep and Shimada, these classes span the middle cohomology, and the intersection form on the primitive part is definite, as predicted by the standard conjecture of Hodge type. We give an elementary proof, which avoids the representation theory of finite unitary groups and uses only the dual polar graph attached to the Hermitian form. It shows that the intersection form on the primitive part is an explicit positive multiple of a Euclidean inner product. The advantage of the new proof is that it also controls $Λ$ $p$-adically. If $X$ is defined over $\F_{q^2}$ and has dimension $2m$, every $p$-adic elementary divisor of the intersection matrix of these subspaces divides $q^m$. Consequently $q^m$ times every algebraic cycle of codimension $m$ is numerically equivalent to an integral combination of them. As a by-product, the standard conjecture of Hodge type holds for varieties finitely covered by Hermitian varieties, and the $p$-adic bound descends along finite morphisms of degree prime to $p$.
CommentsFrom my perspective, this paper is an experiment in itself. It appears that talking with Claude naturally leads to the completion of a paper
