Hasse-Weil Zeta Functions Modulo a Prime

Technical Summary: Hasse-Weil Zeta Functions Modulo a Prime

Problem Statement
The paper addresses the relationship between the Hasse-Weil zeta functions of two separated $F_q$-schemes of finite type, $X$ and $Y$, connected by a finite morphism $\pi: Y \to X$. Specifically, the author investigates the reduction of the zeta function $\zeta(Y, T)$ modulo a prime $r$ (where $r \neq p$, the characteristic of the field) in terms of $\zeta(X, T)$ and the branch locus $Z \subset X$. The primary motivation stems from a question posed by Richard Griffon regarding the conditions under which the numerator of the zeta function of a curve over a finite field is a trinomial. A necessary condition for this is that the numerator is congruent to a trinomial modulo an integer $N > 1$.

Methodology
The paper employs two distinct approaches to establish the main result: a "naive" proof based on Euler-product expansions and a cohomological proof utilizing étale cohomology.

1. Naive Proof (Section 2): This approach analyzes the Euler-product expansion of the zeta function. By examining the fibers of the morphism $\pi$ over closed points $x \in |X|$, the author distinguishes between points in the unramified locus ($X \setminus Z$) and the branch locus ($Z$). Using the properties of the Galois group $G$ (of prime order $r$), inertia groups, and decomposition groups, the author derives congruences for the local factors $L(Y_x, T)$. The proof relies on the observation that $(1 - T^{\deg(x)})^r \equiv (1 - T^{r \deg(x)}) \pmod r$.

2. Cohomological Proof (Sections 3 & 4): This section establishes a general formula for the reduction of a zeta function modulo a prime $\ell \neq p$.

   • Theorem 3.1: The author proves that $\zeta(X, T) \equiv \prod_i \det(1 - \phi T \mid H^i_c(\bar{X}, \mathbb{F}_\ell))^{(-1)^{i+1}} \pmod \ell$. This requires analyzing the relationship between cohomology with coefficients in $\mathbb{Q}_\ell$, $\mathbb{Z}_\ell$, and $\mathbb{F}_\ell$, specifically handling torsion submodules via a linear algebra proposition (Proposition 3.2) concerning determinants of modules over finite rings.
   • Application to $\pi$: In Section 4, the author applies this cohomological formula to the open set $U = X \setminus Z$. By considering the pushforward sheaf $\pi_* \mathbb{F}_r$ and its semisimplification, the author demonstrates that the cohomology of $Y$ over $U$ relates to the cohomology of $X$ over $U$ via a power of $r$. This leads to the congruence $\zeta(V, T) \equiv \zeta(U, T)^r \pmod r$, where $V = \pi^{-1}(U)$.

Key Results
The central result is Theorem 1.1, which states:
Let $F_q$ be a finite field of characteristic $p$, and $\pi: Y \to X$ be a finite $F_q$-morphism of separated $F_q$-schemes of finite type. Let $Z \subset X$ be the branch locus of $\pi$. If $\pi$ is generically Galois with a group $G$ of prime order $r \neq p$, then:
$$\zeta(Y, T) \equiv \zeta(X, T)^r \zeta(Z, T)^{1-r} \pmod r$$

The paper notes a striking feature of this identity: while the analogous identity over $\mathbb{Z}$ generally fails due to weight mismatches, the congruence modulo $r$ holds true.

Applications to Curves
In Section 5, the results are specialized to proper, smooth, geometrically connected curves. The author derives a congruence for the numerators $L(C, T)$ of the zeta functions:
$$L(Y, T) \equiv L(X, T)^r (1-T)^{1-r} (1-qT)^{1-r} \prod_{z \in |Z|} (1 - T^{\deg(z)})^{r-1} \pmod r$$
This section also recovers the Riemann-Hurwitz formula by comparing the degrees of both sides of the congruence.

In Section 6, the paper applies these findings to hyperelliptic ($r=2$) and superelliptic ($r>2$) curves defined by equations of the form $y^r = f(x)$. The results provide a method to determine the reduction modulo $r$ of the numerator of the zeta function for such curves based on the branch locus and the base curve.

Significance and Claims
The paper claims to provide a definitive formula for the mod-$r$ reduction of the zeta function of a Galois cover in terms of the base and the branch locus. The significance lies in:

1. Solving a specific structural question: It offers a necessary condition (congruence to a trinomial) for the numerator of a zeta function to be a trinomial, directly addressing the motivation provided by Richard Griffon.
2. Methodological contribution: It provides both an elementary proof and a cohomological proof, the latter of which establishes a general reduction formula (Theorem 3.1) for zeta functions modulo primes distinct from the characteristic.
3. Clarification of limitations: The author explicitly notes that the identity does not hold over $\mathbb{Z}$ due to weight considerations, highlighting the specific nature of the modular arithmetic context.

The paper does not propose experimental applications or future implications beyond the immediate application to hyperelliptic and superelliptic curves as described.