---
tag: 6ZGF49H346
kind: definition
name: Hasse–Weil zeta function
msc: [14G10, 11G40, 11M41]
defines: 37TNNBQ4PK
uses: [1XWBY4N0VT, 0ZXZ82FTX2, 8MWTEF6462]
sources:
  - url: https://en.wikipedia.org/wiki/Hasse%E2%80%93Weil_zeta_function
---

## Definition

Let $X$ be a scheme of finite type over $\Z$, of Krull dimension $d$.
Let $\abs{X}$ be its set of closed points; the residue field $\kappa(x)$ of each $x \in \abs{X}$ is a [[finite field|0ZXZ82FTX2]], and let $N(x) = \#\kappa(x)$.
The *Hasse–Weil zeta function* of $X$ is
$$\zeta_X(s) = \prod_{x \in \abs{X}} \frac{1}{1 - N(x)^{-s}},$$
an Euler product that converges absolutely for $\mathrm{Re}(s) > d$.

Now let $V$ be a smooth projective variety over $\Q$.
Choose a finite set of primes $S$ and a model $X$ of $V$ that is smooth and projective over $\Z[1/S]$.
For $p \notin S$, let $X_p$ be the fibre of $X$ over $\F_p$, and let
$$Z(X_p, t) = \exp\left( \sum_{m \ge 1} \frac{\#X_p(\F_{p^m})}{m} \, t^m \right).$$
The zeta function of $V$ is the product of the local factors $Z(X_p, p^{-s})$ over $p \notin S$.
Changing the model changes only finitely many factors.

## Remarks

The two forms agree: the local factor $Z(X_p, p^{-s})$ is the product of $(1 - N(x)^{-s})^{-1}$ over the closed points $x$ of $X$ lying over $p$.

Example: for $X = \mathrm{Spec}\, \Z$, the closed points are the primes $p$ with $N(p) = p$, so $\zeta_X(s) = \prod_p (1 - p^{-s})^{-1} = \zeta(s)$, the [[Riemann zeta function|1XWBY4N0VT]], which converges for $\mathrm{Re}(s) > 1 = d$.
For $X = \mathbb{A}^1_{\Z}$, one finds $\zeta_X(s) = \zeta(s - 1)$, since $\#\mathbb{A}^1(\F_{p^m}) = p^m$.

Non-example: the [[Dirichlet series|8MWTEF6462]] $\sum_{n \ge 1} (-1)^{n} n^{-s}$ is not of this form, since its coefficients take negative values, while expanding an Euler product of factors $(1 - N(x)^{-s})^{-1}$ gives nonnegative integer coefficients.

For an elliptic curve $E$ over $\Q$ with a good model away from its conductor, the local factor at a good prime $p$ is $\dfrac{1 - a_p p^{-s} + p^{1 - 2s}}{(1 - p^{-s})(1 - p^{1-s})}$.
The Hasse–Weil conjecture predicts meromorphic continuation to $\C$ and a functional equation; it is known for elliptic curves over $\Q$ by the modularity theorem.