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.



Motivating the coefficient field of $\ell$-adic cohomology
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.


