论文标题
尊重多重性的算术zeta函数
An arithmetic zeta function respecting multiplicities
论文作者
论文摘要
在本文中，我们研究了算术Zeta函数$$ \ MATHSCR {Z} _ {\ MATHCAL {X}}}（s）= \ prod_p \ prod _ {\ ordack {\ ordack {x \ in \ Mathcal {x} \ frac {1} {1- |κ（x）|^{ - s}}} \ big）^{\ Mathfrak {\ Mathfrak {m} _ {p}（x）} $$与方案$ \ Mathcal {x} $ cons $ \ \ mathbb {z} $ bive there $ bl {$κ（x）$ bive consecy $ \ mathcal {x} $关联的$κ（x） $ \ mathfrak {m} _ {p}（x）$ $ x $ in $ \ mathcal {x} _p $的$ x $。如果在有限字段上定义了$ \ Mathcal {x} $，则$ \ Mathscr {z} _ {\ Mathcal {x}} $自然出现在与多数次数计数的上下文中。我们证明$ \ Mathscr {z} _ {\ Mathcal {x}} $允许持续到$ \ {s \ in \ Mathbb {C} \ colon \ colon \ Mathrm {re} re}（re}（s）> \ mathrm {dim {dim {dim} {$ nater oftery in \ colon \ colon \ mathrm {re}在$ s = \ mathrm {dim}（\ mathcal {x}）$。最后，我们将$ \ mathscr {z} _ {\ mathcal {x}} $与zeta函数$ζ_F$编码多项式$ f $的剩余分解模式。
In this paper, we study the arithmetic zeta function $$\mathscr{Z}_{\mathcal{X}}(s) = \prod_p \prod_{\substack{x \in \mathcal{X}_p \\ \text{closed}}} \Big( \frac{1}{1-|κ(x)|^{-s}} \Big)^{\mathfrak{m}_{p}(x)}$$ associated to a scheme $\mathcal{X}$ of finite type over $\mathbb{Z}$, where $κ(x)$ denotes the residue field and $\mathfrak{m}_{p}(x)$ the multiplicity of $x$ in $\mathcal{X}_p$. If $\mathcal{X}$ is defined over a finite field, then $\mathscr{Z}_{\mathcal{X}}$ appears naturally in the context of point counting with multiplicities. We prove that $\mathscr{Z}_{\mathcal{X}}$ admits a meromorphic continuation to $\{s \in \mathbb{C} \colon \mathrm{Re}(s) > \mathrm{dim}(\mathcal{X})-1/2\}$ and determine the order of its pole at $s = \mathrm{dim}(\mathcal{X})$. Finally, we relate $\mathscr{Z}_{\mathcal{X}}$ to a zeta function $ζ_f$ encoding the residual factorization patterns of a polynomial $f$.
