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Added reference to comments on point counts.
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Andreas Holmstrom
Andreas Holmstrom
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Let $X$ be a regular scheme, flat and of finite type over $Spec(\mathbb{Z})$ (add "projective" if you want). Then the Hasse-Weil zeta function of $X$ is defined as a product over all prime numbers of certain local factors which are rational functions in $p^{-s}$. The local factor at $p$ is the zeta function of the fiber $X_p$, which is a variety over the finite field $\mathbb{F}_p$.

For all but finitely many primes, these local factors should have "similar shape", in some sense. For example, for an elliptic curve, and a good prime $p$, the numerator is a polynomial with coefficients $(1, a_p, p)$, i.e. all these numerators are exactly the same, except of course that the prime $p$ varies. For the denominators the situation is similar.

If we take a higher-genus curve, or a higher-dimensional scheme, the patterns of the local zeta function coefficients should also in some sense be "uniform in p". But what exactly is the statement in the general case? In what precise sense are the local factors "the same"?

EDIT: I added some (hopefully clarifying) comments related to point counts under the question as well as under ACL's answer.

Let $X$ be a regular scheme, flat and of finite type over $Spec(\mathbb{Z})$ (add "projective" if you want). Then the Hasse-Weil zeta function of $X$ is defined as a product over all prime numbers of certain local factors which are rational functions in $p^{-s}$. The local factor at $p$ is the zeta function of the fiber $X_p$, which is a variety over the finite field $\mathbb{F}_p$.

For all but finitely many primes, these local factors should have "similar shape", in some sense. For example, for an elliptic curve, and a good prime $p$, the numerator is a polynomial with coefficients $(1, a_p, p)$, i.e. all these numerators are exactly the same, except of course that the prime $p$ varies. For the denominators the situation is similar.

If we take a higher-genus curve, or a higher-dimensional scheme, the patterns of the local zeta function coefficients should also in some sense be "uniform in p". But what exactly is the statement in the general case? In what precise sense are the local factors "the same"?

Let $X$ be a regular scheme, flat and of finite type over $Spec(\mathbb{Z})$ (add "projective" if you want). Then the Hasse-Weil zeta function of $X$ is defined as a product over all prime numbers of certain local factors which are rational functions in $p^{-s}$. The local factor at $p$ is the zeta function of the fiber $X_p$, which is a variety over the finite field $\mathbb{F}_p$.

For all but finitely many primes, these local factors should have "similar shape", in some sense. For example, for an elliptic curve, and a good prime $p$, the numerator is a polynomial with coefficients $(1, a_p, p)$, i.e. all these numerators are exactly the same, except of course that the prime $p$ varies. For the denominators the situation is similar.

If we take a higher-genus curve, or a higher-dimensional scheme, the patterns of the local zeta function coefficients should also in some sense be "uniform in p". But what exactly is the statement in the general case? In what precise sense are the local factors "the same"?

EDIT: I added some (hopefully clarifying) comments related to point counts under the question as well as under ACL's answer.

Source Link
Andreas Holmstrom
Andreas Holmstrom
  • 5.6k
  • 5
  • 41
  • 62

Local factors of Hasse-Weil zeta function - what do they have in common?

Let $X$ be a regular scheme, flat and of finite type over $Spec(\mathbb{Z})$ (add "projective" if you want). Then the Hasse-Weil zeta function of $X$ is defined as a product over all prime numbers of certain local factors which are rational functions in $p^{-s}$. The local factor at $p$ is the zeta function of the fiber $X_p$, which is a variety over the finite field $\mathbb{F}_p$.

For all but finitely many primes, these local factors should have "similar shape", in some sense. For example, for an elliptic curve, and a good prime $p$, the numerator is a polynomial with coefficients $(1, a_p, p)$, i.e. all these numerators are exactly the same, except of course that the prime $p$ varies. For the denominators the situation is similar.

If we take a higher-genus curve, or a higher-dimensional scheme, the patterns of the local zeta function coefficients should also in some sense be "uniform in p". But what exactly is the statement in the general case? In what precise sense are the local factors "the same"?