The Riemann hypothesis for finite fields can be stated as follows: take a smooth projective variety X of finite type over the finite field $\mathbb{F}\_q$ for some $q=p^n$. Then the eigenvalues $\alpha\_j$ of the action of the Frobenius automorphism on the $i$th $\ell$-adic étale cohomology (are algebraic numbers and) have norm $q^{i/2}$. This is part of the general philosophy that led to the proof of the Weil conjectures. What I don't understand, and pardon me for saying this, is why we care about these eigenvalues.


As a homotopy theorist, I've found the other three conjectures (rationality, Betti numbers, and the functional equation) to be helpful in understanding zeta functions from a geometric perspective. Together, they describe the relationship between the combinatorics and the cohomology of a variety. The Riemann hypothesis, however, doesn't seem to admit a direct interpretation of this kind; it's unclear what role the $\alpha$s play in the analogy. Do they have an interpretation as the arithmetic version of a classical geometric/topological object, like how the degrees are Betti numbers? If not, how should I understand them?



Why do we care about the eigenvalues of the Frobenius map?
The Riemann hypothesis is very important for the relationship between the cohomology and combinatorics of the variety.


First, the Riemann hypothesis lets us read off the Betti numbers from the point counts over finite fields, i.e. the $i$'th Betti number is the number of zeroes/poles of $$e^{ \sum\_j \# X(\mathbb F\_q^j) u^j / j }$$ of absolute value $q^{-i/2}$.


Without the Riemann hypothesis, and with just the other Weil conjectures, it's not possible to calculate the Betti numbers in this way, because you can't distinguish which zeroes or poles are coming from which $P\_i$s or, worse, rule out the case that zeroes and poles will cancel. Without the Riemann hypothesis, one can only calculate the Euler characteristic.


Second, the Riemann hypothesis lets us get information about point counts over finite fields from the Betti numbers. The simplest of these is the upper bound $$|X(\mathbb F\_q)| \leq \sum\_{i=0}^{2n} \dim H^i(X) q^{i/2}.$$ Without the Riemann hypothesis, only much weaker results of this form could be proven (maybe one could replace $q^{i/2} $ with $q^{ \max(i,n)}$ or something like that). Without even a crude bound, even knowing exactly the Betti numbers won't typically rule out any particular value for the number of points over a given field.


I would even say this is much more direct than the relationship between geometry and combinatorics obtained from the remaining Weil conjectures.




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In terms of an analogue in classical geometry / topology, the obvious thing would be the eigenvalues of the action of a map on the cohomology! Of course, one usually doesn't have an a priori exact formula for the absolute value of the eigenvalues, but if one did, it would certainly be useful for understanding the fixed points of the map.


So the Riemann hypothesis is a new phenomenon that doesn't have an analogue in topology (except for Serre's analogue of the Weil conjectures for Kahler manifolds), but the eigenvalues of operators acting on cohomology were a pre-existing notion. Lefschetz certainly wasn't thinking about the Frobenius when he proved his original fixed point formula!


Maybe one should mention also that the eigenvalues of the mapping class of a surface acting on its cohomology give you information on where that mapping class sits in the Nielsen-Thurston classification.




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There is one aspect to classical analogues that I think deserves mentioning because it's of great importance:


The Riemann hypothesis in the Weil conjectures tells us that *calculating the high-degree (compactly-supported) or low-degree (if the variety is smooth) cohomology groups of a variety in topology is analogous to obtaining an approximate estimate for the number of points in arithmetic*.


This is the starting point for deep connections between stable homology and other topological methods for calculating the low-degree cohomology groups without necessarily calculating every cohomology group, and analytic number theory or other fields where quantities are calculated approximately!


So RH is not an analogue of anything classical in topology but it tells us what the analogues of some classical statements in topology are.


