Let A be an abelian fourfold in characteristic p . We prove the standard conjecture of Hodge type for A , namely that the intersection product $$\begin{aligned} {\mathcal {Z}}^2_{\mathrm {num}}(A)_{{\mathbb {Q}}}\times {\mathcal {Z}}_{\mathrm {num}}^2(A)_{{\mathbb {Q}}} \longrightarrow {\mathbb {Q}}\end{aligned}$$ Z num 2 ( A ) Q × Z num 2 ( A ) Q ⟶ Q is of signature $$(\rho _2 - \rho _1 +1; \rho _1 - 1)$$ ( ρ 2 - ρ 1 + 1 ; ρ 1 - 1 ) , with $$\rho _n=\dim {\mathcal {Z}}_{\mathrm {num}}^n(A)_{{\mathbb {Q}}}.$$ ρ n = dim Z num n ( A ) Q . (Equivalently, it is positive definite when restricted to primitive classes for any choice of the polarization.) The approach consists in reformulating this question into a p -adic problem and then using p -adic Hodge theory to solve it. By combining this result with a theorem of Clozel we deduce that numerical equivalence on A coincides with $$\ell $$ ℓ -adic homological equivalence on A for infinitely many prime numbers $$\ell $$ ℓ . Hence, what is missing among the standard conjectures for abelian fourfolds is $$\ell $$ ℓ -independency of $$\ell $$ ℓ -adic homological equivalence.

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阿贝尔四重的标准猜想

设 A 是特征 p 的四倍阿贝尔。我们证明了 A 的 Hodge 类型的标准猜想，即交积 $$\begin{aligned} {\mathcal {Z}}^2_{\mathrm {num}}(A)_{{\mathbb {Q} }}\times {\mathcal {Z}}_{\mathrm {num}}^2(A)_{{\mathbb {Q}}} \longrightarrow {\mathbb {Q}}\end{aligned}$$ Z num 2 ( A ) Q × Z num 2 ( A ) Q ⟶ Q 的签名为 $$(\rho _2 - \rho _1 +1; \rho _1 - 1)$$ ( ρ 2 - ρ 1 + 1 ; ρ 1 - 1 ) , $$\rho _n=\dim {\mathcal {Z}}_{\mathrm {num}}^n(A)_{{\mathbb {Q}}}.$$ ρ n = 昏暗 Z num n ( A ) Q 。（等效地，当限制为任何极化选择的原始类时，它是正定的。）该方法包括将这个问题重新表述为 ap -adic 问题，然后使用 p -adic Hodge 理论来解决它。通过将这个结果与 Clozel 定理相结合，我们推导出 A 上的数值等价性与 A 上的 $$\ell $$ ℓ -adic 同调等价性对无穷多个素数 $$\ell $$ ℓ 重合。因此，在阿贝尔四重的标准猜想中缺少的是 $$\ell $$ ℓ - $$\ell $$ ℓ -adic 同调等价的独立性。