Let $X\_0$ be a smooth projective variety over a finite field $\mathbb{F}\_q$. Let $X$ be the corresponding variety over the algebraic closure $\bar{\mathbb{F}}\_q$. Let $Fr\_q\colon X\to X$ be the geometric Frobenius.


**Is it true that the eigenvalues of $Fr\_q$ on $H^i(X,\mathbb{Q}\_l)$ ($l\ne char(\mathbb{F}\_q)$) are algebraic integers independent of $l$? Is the same true if $X\_0$ is not necessarily smooth and projective?**


A reference would be helpful.


**Remark.** I know that in the smooth projective case the eigenvalues are algebraic numbers (I do not know if they are integers).


Sorry if this question is well known to experts, I am not from this field.



Eigenvalues of Frobenius in $l$-adic cohomology
This was proven by Deligne in the smooth projective case [Weil I, Thm. I.6], and later in the smooth proper case [Weil II, Cor. 3.3.9].


In general (already for smooth *quasi*-projective varieties), we don't even know whether $\dim H^i(X\_{\text{ét}},\mathbf Q\_\ell)$ is independent of $\ell$. See for instance [Katz, p. 28, 2(a)] or [Illusie, 3.5(c)]. I also wrote a paper on this [vDdB] that may contain some more references.




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**References.**


[Weil I] P. Deligne, [*La conjecture de Weil. I*](http://www.numdam.org/item/PMIHES_1974__43__273_0/). Publ. Math. Inst. Hautes Étud. Sci. **43**, p. 273-307 (1973). [ZBL0287.14001](https://zbmath.org/?q=an:0287.14001).


[Weil II] P. Deligne, [*La conjecture de Weil. II*](http://www.numdam.org/item/PMIHES_1980__52__137_0/), Publ. Math. Inst. Hautes Étud. Sci. **52**, p. 137-252 (1980). [ZBL0456.14014](https://zbmath.org/?q=an:0456.14014).


[Katz] N. M. Katz, *Review of $\ell$-adic cohomology*. In: [Motives](http://dx.doi.org/10.1090/pspum/055.1). Proc. Symp. Pure Math. **55**.1, p. 21-30 (1994). [ZBL0817.14006](https://zbmath.org/?q=an:0817.14006).


[Illusie] L. Illusie, [*Miscellany on traces in $\ell$-adic cohomology: a survey*](https://doi.org/10.1007/s11537-006-0504-3). Jpn. J. Math. (3) **1**.1, p. 107-136 (2006). [ZBL1156.14309](https://zbmath.org/?q=an:1156.14309).


[vDdB] R. van Dobben de Bruyn, [*The equivalence of several conjectures on independence of $\ell$*](https://doi.org/10.46298/epiga.2020.volume4.5570). Épijournal de Géom. Algébr. **4**, Article No. 16 (2020). [ZBL1460.14053](https://zbmath.org/?q=an:1460.14053).


