Why do we care about the eigenvalues of the Frobenius map?

Question

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?

Answer 1

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.

Answer 2

Here are a few different uses of knowing how large the eigenvalues are as complex numbers.

Application 1: Bounding exponential sums. Many classical exponential sums can be interpreted essentially as a trace (sum) of such eigenvalues, which is a totally different set of terms being added than the original exponential sum: an initial exponential sum of $p$ terms over the field $\mathbf F_p$ might be a trace of Frobenius on a low-dimensional space, for instance of dimension $2$ so the exponential sum of $p$ terms equals a sum of two terms $\alpha_p + \beta_p$. Then RH lets you bound the exponential sum, like in Hasse’s bound $|a_p| = |\alpha_p + \beta_p| \leq |\alpha_p| + |\beta_p| \leq 2\sqrt{p}$ for an elliptic curve over $\mathbf F_p$. While RH for curves helps you get sharp bounds for exponential sums in one variable, RH for higher-dimensional varieties leads to sharp bounds on exponential sums in several variables.

Application 2: Determining sharper half-planes of convergence for Euler products. That the Euler product defining the $L$-function of an elliptic curve over $\mathbf Q$ converges absolutely for ${\rm Re}(s) > 3/2$ is due to the bound $|a_p| \leq 2\sqrt{p}$ for primes $p$ of good reduction, which comes from $|\alpha_p| = |\beta_p| = \sqrt{p}$. If we didn't have that information and only had the cruder estimate $|a_p| \leq p$ from the description of $a_p$ as a sum over $\mathbf F_p$, then we could only say the Euler product converges on the smaller right half-plane ${\rm Re}(s) > 2$. This application can be extended to the Euler products defining more general Hasse-Weil $L$-functions of varieties over number fields, of which elliptic curves over $\mathbf Q$ are a special case.

Application 3: Delinge’s proof of the Ramanujan-Petersson conjecture for modular forms associated to congruence subgroups of ${\rm SL}_2(\mathbf Z)$ as a consequence of his proof of RH and prior work of Eichler, Shimura, Kuga, and others. In particular, Ramanujan’s conjecture that for the coefficients of the weight $12$ cusp form $\Delta(q) = \sum_{n\geq 1} \tau(n)q^n$ we have $|\tau(p)| \leq 2p^{11/2}$ for all primes $p$ relies on RH for varieties over finite fields. Finding the relevant variety over $\mathbf F_p$ for this application is not easy.

Whether you think these are worthwhile may be a matter of taste, but each of them is definitely interesting to some people. The first application (improving bounds on exponential sums) was historically one of the motivations for proving RH over finite fields, at least for curves. See the last section of Roquette's first paper on the historical development of RH in characteristic $p$ here and section 3 (and much more) of his second paper here. If you replace "rv2" in the URL of the second paper with "rv3" and "rv4" you'll see parts 3 and 4. You don't have to have a definition of the zeta-function of a variety to care about exponential sums and how large they can be: many exponential sums (named after Gauss, Jacobi, Kloosterman, et al.) were of interest independently of having a definition of zeta-functions of varieties over finite fields, and the account here shows how counting questions about quadratic residue patterns from the late 19th and early 20th centuries lead naturally to interest in bounding exponential sums.

Two more references: Dieudonne’s article "On the History of the Weil Conjectures", either in The Mathematical Intelligencer 10 (1975), 7-21 or in reprinted form at the start of Freitag and Kiehl's Etale Cohomology and the Weil Conjecture[s] and Katz’s survey paper “An overview of Deligne’s proof…” in the Proc. Symp. Pure Math (volume 28) on Hilbert Problems published by the AMS in 1976.

Answer 3

All of the other answers give good reasons why the eigenvalues and their absolute values are important, but it should be noted that the eigenvalues can be used to give an exact point count via the fixed point formula. Thus if $X/\mathbb F_q$ is a smooth projective variety and if we denote the eigenvalues of $\Phi_q$ on the $i$th etale cohomology by $\lambda_{ij}$, then for all $n\ge1$, $$ \#X(\mathbb F_{q^n}) = \sum_{i=0}^{2d} (-1)^i \sum_{j=1}^{b_i} \lambda_{ij}^n. \tag{$*$} $$ The main term in $(*)$ is $q^d$, and the Riemann hypothesis estimates on the other eigenvalues give $$ \#X(\mathbb F_{q^n}) = q^{dn} + O\left(q^{(d-\frac12)n}\right), \tag{$**$} $$ where the big-$O$ constant depends only on the Betti numbers. The formula $(*)$ with the precise values of the absolute values of the terms is the starting point for a huge amount of research concerning the points on varieties over finite fields.

Addendum: As KConrad notes, the estimate $(**)$ was proven earlier by Lang and Weil [1] with constants that depend fairly explicitly on the geometry of the embedding of the variety into projective space. However, the estimates obtained are weaker than that which comes from $(*)$, Deligne's theorem, and the triangle inequality, to whit: $$ \#X(\mathbb F_{q^n}) \le q^{dn} + \sum_{i=1}^{2d} b_i\cdot q^{ni/2}. $$ For example, if $H^1_{\text{et}}(X/\mathbb F_q,\mathbb Q_\ell)=0$, i.e., if $b_1=0$, then the error improves to $O(q^{(d-1)n})$.

[1] Lang, Serge; Weil, André, Number of points of varieties in finite fields, Am. J. Math. 76, 819-827 (1954). ZBL0058.27202.

Answer 4

I'm surprised that the great answers so far do not mention weights at all. One consequence of Deligne's theorem is that the zeta function of a smooth projective variety knows the Betti numbers: $h^i(X)$ is the number of zeroes ($i$ odd) or poles ($i$ even) of $Z(X,t)$ with absolute value $q^{i/2}$.

On varieties that are not smooth proper (i.e. singular or open), this is not quite right; for instance if $X = \mathbf A^1\setminus\{0\}$, then $\lvert X(\mathbf F_q)\rvert = q-1$, but $h^i_c(X) = 1$ for $i\in\{1,2\}$. This is saying that $H^1_c(X,\mathbf Q_\ell)$ truly "comes from an $H^0_c$" (i.e. has weight $0$), even though it looks like an $H^1$. (Of course in this case there is an easy explanation in terms of the long exact sequence for compactly supported cohomology for the inclusion $X \subseteq \mathbf P^1$.)

This led Deligne to define mixed Hodge structures on the cohomology of singular or open complex algebraic varieties. This analogy is explained in Théorie de Hodge I (a writeup of an ICM address), and worked out in parts II and III (actual journal articles). So there is an analogue of weight theory in complex algebraic geometry, except that the analogy historically went the other way around!

The analogy between the Galois action on $H^i(X_{\bar k},\mathbf Q_\ell)$ and the pure Hodge structure on $H^i(X_{\mathbf C},\mathbf Z)$ (say for smooth projective varieties) is not merely philosophical. In $p$-adic geometry, they closely interact in ways described by $p$-adic Hodge theory, and over number fields they are conjecturally linked by the Mumford–Tate conjecture. In both cases, the Galois groups are substantially larger than $\operatorname{Gal}(\bar{\mathbf F}_q/\mathbf F_q)$, but the finite field case is an important case to start with.