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Let $C$ be a smooth projective curve over a finite field $\mathbb F_q$, $q$ is a power of the characteristic $p$. It is well-known that if $\alpha$ is an eigenvalue of Frobenius acting on $H^1_{et}(C,\mathbb Q_{\ell})$ (that is, on $\ell$-adic etale cohomology, where $\ell$ is a prime distinct from $p$) then $\alpha$ is an algebraic integer and $\alpha \overline \alpha=q$ (i.e., $\alpha$ is a Weil number).

My question is: what can one say about the maximal rational $v$ such that $\alpha\cdot q^{-v}$ is an algebraic integer (certainly, this can be re-formulated in terms of valuations of $\alpha$ at the primes lying above $p$; hence a rational maximum $v$ exists indeed). Can $v$ be distinct from $0$ and $1/2$?

Actually, I am interested about the generalization of this question to all cohomology spaces of arbitrary smooth projective varieties over $\mathbb F_q$; yet it appears that Honda's "Isogeny classes of abelian varieties over finite fields" (along with the Rieman Hypothesis over finite fields and other properties of etale cohomology) reduce this general question to the case of curves.

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Every value $c \in \big[0,\tfrac{1}{2}\big] \cap \mathbf Q$ can occur as the smallest slope of an abelian variety over $\mathbf F_q$; see the corollary below.

What Honda actually proves [Hon68] (see [Mil94, Prop. 2.6] for a motivic reinterpretation) is:

Theorem (Honda). Let $q$ be a power of a prime $p$. Then the map \begin{align*} \frac{\{\text{simple abelian varieties } A \text{ over } \mathbf F_q\}}{\{\mathbf F_q\text{-isogeny}\}} &\to \frac{\{q\text{-Weil numbers}\}}{\text{conjugacy}}\\ A &\mapsto \operatorname{Frob}_A \end{align*} is a bijection.

In other words, every $q$-Weil number (of weight $1$) is realised inside some abelian variety; in particular inside some smooth projective curve (cut by smooth hyperplanes).

So the only question is: what are the $q$-Weil numbers? This is a purely number theoretic question. If you only want to know the valuation¹ of $\alpha$ and all its conjugates, it is not so hard to come up with examples.

Lemma. Let $a, b \in \mathbf Z$ be coprime with $0\leq a \leq \tfrac{b}{2}$, and let $\alpha$ be a root of $$f(x) = x^{2b} + q^ax^b + q^b.$$ Then $\alpha$ is a $q$-Weil number with slopes $$\big\{\underbrace{\tfrac{a}{b},\ldots,\tfrac{a}{b}}_b,\underbrace{\tfrac{b-a}{b},\ldots,\tfrac{b-a}{b}}_b\big\}.$$

Proof. If $g(x) = x^2 + q^ax + q^b$, then $\beta = \alpha^b$ is a root of $g$. Note that $g$ is irreducible over $\mathbf Q$ (even over $\mathbf R$) since $$\Delta = q^{2a} - 4q^b < 0.$$ Hence $\beta\bar\beta = q^b$ (the constant term of $g$), so $\alpha\bar\alpha = q$. Clearly the Newton polygon of $f$ has slopes $\tfrac{a}{b}$ and $\tfrac{b-a}{a}$, both with multiplicity $b$. $\square$

In fact it's not hard to see that $f$ is irreducible when $q = p$, using Newton polygons and irreducibility of $g$, but we don't need this.

Corollary. For every $c \in \big[0,\tfrac{1}{2}\big] \cap \mathbf Q$, there exists a simple abelian variety $A$ over $\mathbf F_q$ such that the smallest slope of $H^1_{\operatorname{\acute et}}(A_{\bar{\mathbf F}_q},\mathbf Q_\ell)$ is $c$.

Proof. Write $c = \frac{a}{b}$ with $a$ and $b$ coprime, take $\alpha$ as in the lemma, and apply Honda's theorem to get a simple abelian variety over $\mathbf F_q$ with slopes the conjugates of $\alpha$. $\square$

Remark. The question which precise set of slopes can occur on smooth projective curves is a very difficult one, and this is an active area of study. For example, it is not known if for every $(g,p)$ there exists a curve $C$ of genus $g$ in characteristic $p$ such that all slopes are $\tfrac{1}{2}$ (i.e. $C$ is supersingular). If you only care about one eigenvalue and its conjugates, you can reduce to the answer above.


¹ We choose a prime above $p$ (equivalently, pick an embedding $\bar{\mathbf Q} \hookrightarrow \bar{\mathbf Q}_p$) and normalise the valuation so that $v(q) = 1$. Then studying the valuations of $\alpha$ at primes above $p$ has become studying the valuation of the conjugates of $\alpha$, which are called the slopes of $\alpha$.


References.

[Hon68] T. Honda, Isogeny classes of abelian varieties over finite fields. J. Math. Soc. Japan 20, p. 83-95 (1968). ZBL0203.53302.

[Mil94] J. S. Milne, Motives over finite fields. Motives (Seattle, WA). Proc. Symp. Pure Math. 55.1, p. 401-459 (1994). ZBL0811.14018.

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    $\begingroup$ Thank you very much! I was actually interested in the effectivity of motives (and their tensor products). It appears that in characteristic $0$ only "integral effectivity" is possible, whereas the effectivity of motives in characteristic $p$ can be "fractional". $\endgroup$ Apr 4 '20 at 7:31
  • $\begingroup$ @MikhailBondarko: ah, interesting. I think I agree with you, and I had never thought of that! $\endgroup$ Apr 4 '20 at 18:46

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