It was already known to Weil that a sufficiently reasonable cohomology theory for algebraic varieties over $\mathbb{F}_p$ would allow for a possible solution to the Weil conjectures.

It was also understood that such a cohomology theory could not take values in vector spaces over either the rational numbers $\mathbb{Q}$ or the $p$-adic numbers $\mathbb{Q}_p$. The classical argument that I know is that the cohomology of a supersingular elliptic curve should admit an action by a quaternion algebra structure of the endomorphism group of the elliptic curve, and representation theory rules out such an action. (Side question: Due to whom is this reasoning, and was it known to Weil's generation?)

This reasoning does not, however, rule out the possibility of cohomology with coefficients in $\mathbb{Q}_\ell$ with $\ell \neq p$, and indeed $\ell$-adic cohomology would end up being found.

Question. Were there any other historical reasons to believe that looking at cohomology valued in $\mathbb{Q}_\ell$-vector spaces would lead to something fruitful?

N.B. This question was posted on MathSE, but I got advised to post it here instead.

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    $\begingroup$ The argument with supersingular elliptic curves and quaternion algebras is due to Serre, I believe. $\endgroup$ – Piotr Achinger Jan 30 at 14:03
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    $\begingroup$ (See 1.7, p. 315 in Grothendieck's "Crystals and the de Rham cohomology of schemes", "As Serre has pointed out...") $\endgroup$ – Piotr Achinger Jan 30 at 14:31
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    $\begingroup$ It is not true that etalé cohomology with coefficients in $\mathbb{Q}_\ell$ is a "good" cohomology. $\ell$-adic cohomology is something more involved that just "cohomology with coefficients": its is the projective limit of some with finite coefficients (and tensoring with $\mathbb{Q}$). $\endgroup$ – Xarles Jan 30 at 14:53
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    $\begingroup$ @Xarles That is of course what I meant. It takes values in $\mathbb{Q}_\ell$-modules, anyway. I'll edit the question accordingly. $\endgroup$ – Mr. Palomar Jan 30 at 15:55

One historical reason for considering $\ell$-adic cohomology, not completely disconnected from the example you introduce, is that for a curve over a field, we get a natural Galois representation by taking the $\ell$-adic Tate module of the Jacobian (i.e., the projective limit of $\ell$-power torsion). Furthermore, if such a curve is defined over a subfield of the complex numbers, then the rank of the Tate module is equal to the rank of the classical degree 1 cohomology of the complexified curve. We now know that the Tate module is naturally dual to étale cohomology in degree 1.

One might then reasonably hope for a similar relationship between the higher degree classical cohomology of higher dimensional varieties and certain $\ell$-adic Galois representations naturally attached to the variety.

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