AI 中文总结
本文利用δ-几何理论，对任意阿贝尔安德森模E构造了带霍奇-平克结构的典范z-等晶体，证明其与de Rham上同调的相容关系及同构，还验证了Drinfeld模、Carlitz模的相关性质。
AI 中文摘要
在本文中，利用δ-几何理论，我们对任意阿贝尔安德森模E构造了一个具有霍奇-平克结构的典范z-等晶体（H_δ(E), f^*）。H_δ(E)上的霍奇-平克结构诱导出一个自然滤过（H_δ(E) ⊃ X_prim(E) ⊃ {0}），X_prim(E)的元素由与E相关的本原δ特征表示。我们建立了从H_δ(E)到相伴 de Rham 上同调模H^*_{dR}(E)的自然态射，该态射与前述滤过及H^*_{dR}(E)上的经典霍奇滤过（H^*_{dR}(E) ⊃ Lie(E)^* ⊃ {0}）严格相容。此外，我们证明该映射诱导出X_prim(E)与Lie(E)^*之间的同构，因此该同构将E的不变微分解释为E的本原δ特征。当E为Drinfeld模时，我们证明所构造的z-等晶体H_δ(E)是弱可容许的，因此正特征下的Fontaine函子类比将结晶z-进伽罗瓦表示与δ-几何对象H_δ(E)关联起来。当E为Carlitz模时，我们证明与H_δ(E)关联的伽罗瓦表示确实是来自Tate模的常用表示。
英文摘要
In this article, using the theory of $δ$-geometry, we construct a canonical $z$-isocrystal $(\mathbf{H}_δ(E), \mathfrak{f}^*)$ admitting a Hodge-Pink structure for any abelian Anderson module $E$. The Hodge-Pink structure on $\mathbf{H}_δ(E)$ induces a natural filtration $(\mathbf{H}_δ(E) \supset \mathbf{X}_{\mathrm{prim}}(E) \supset \{0\})$. The elements of $\mathbf{X}_{\mathrm{prim}}(E)$ are represented by primitive delta characters associated to $E$. We establish a natural morphism from $\mathbf{H}_δ(E)$ to the associated de Rham cohomology module $\mathbf{H}^*_{\mathrm{dR}}(E)$, which is strictly compatible with the aforementioned filtration and the classical Hodge filtration $(\mathbf{H}^{*}_{\mathrm{dR}}(E)\supset {\mathrm{Lie}(E)^{*}}\supset \{0\})$ on $\mathbf{H}^*_{\mathrm{dR}}(E)$. Moreover, we show that the map induces an isomorphism between $\mathbf{X}_{\mathrm{prim}}(E)$ and $\mathrm{Lie}(E)^*$. Hence our isomorphism provides an interesting interpretation of the invariant differentials of $E$ as primitive delta characters of $E$. Furthermore, when $E$ is a Drinfeld module, we show that the constructed $z$-isocrystal $\mathbf{H}_δ(E)$ is weakly admissible. Consequently, the positive equal characteristic analogue of the Fontaine functor associates a crystalline $z$-adic Galois representation to the $δ$-geometric object $\mathbf{H}_δ(E)$. In the case, when $E$ is the Carlitz module, we show that the Galois representation associated to $\mathbf{H}_δ(E)$ is indeed the usual one coming from the Tate module.
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