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.