Actually I am not sure this is a legitimate question on MO. In April and June of this year Serre gave two talks on the same title "linear representations and the number of points mod p", one in ETH Number theory Days Zurich, another during Prof. Gross's birthday conference in Boston. Unfortunately I was in neither of those, nor could I find another reference about this talk online.


In the proof of Weil conjecture for curves, say an elliptic curve $E$ over $\mathbf{Q}$ has good reduction mod $p$, then the characteristic polynomial of the Frobenius operator on the Tate module for $E$ mod $p$ will essentially give us the number of points on the curve in $\mathbf{F}\_{p^n}$. So I would say this is an obvious example of relations between *two*-dimensional representation and number of points. But I wonder if Serre has more. E.g. (tangentially related) [Here](http://www.math.harvard.edu/conferences/gross_10/panel/panel.pdf) Mazur was interested in the **Chebyshev bias** (which is usually a quite analytic phenomenon) among the number of points corresponding to different $p$ (which, by the way, has few references also), and I'd like to know if the representation side could shed some light on this. 


Of course since I didn't attend the talk, Serre could be talking about totally different things. I would greatly appreciate it if anyone attended the talk/have seen such notes/heard about this circle of ideas share some comments on this. Thanks!



Looking for reference on Serre's talk "linear rep and number of points mod p"
At the Dick Gross conference, Serre went over what he called "missing exercises from SGA 4.5". Basically he used the relation between the number of points mod $p$ on a variety and eigenvalues of Frobenius at $p$ to make statements that were less obvious in one situation, but pretty clear in the other (one side being points mod $p$, the other side being statements about functions on topological groups). Here's the dictionary: Let $X$ be a separated finite type scheme over $\mathbf{Z}$ and let $N\_X(p^e)=\mathrm{card}(X(\mathbf{F}\_{p^e}))$. On the one hand, one has that


* $N\_X(p^e)=\sum (-1)^i\mathrm{Tr}\left(\mathrm{Fr}\_p^e|H^i\_c(\overline{X},\mathbf{Q}\_\ell)\right)$ for all $p\geq p\_0$ and all $e\geq1$.


On the other hand,


* let $G=\mathrm{Gal}(\mathbf{Q}\_S/\mathbf{Q})$, where $\mathbf{Q}\_S$ is the maximal extension of $\mathbf{Q}$ unramified outside a finite set of primes $S$. Consider the virtual character $a$ of $\sum(-1)^iH^i\_c\left(\overline{X},\mathbf{Q}\_\ell\right)$.


Here are some situations he looked at:


1. If $N\_X(p)=N\_{X^\prime}(p)$ for a set of primes of density 1 then $N\_X(p^e)=N\_{X^\prime}(p^e)$ for all $p\geq p\_0$ and all $e\geq1$. (Under the dictionary, this is simply the statement that if $G$ is a topological group, $K$ is a topological field, and $a, a^\prime$ are two continuous functions $G\rightarrow K$ that agree on a dense subset, then they are equal.)
2. Suppose $|N\_X(p)-N\_{X^\prime}(p)|$ is bounded for a set of primes of density 1. Then


(i) $|N\_X(p^e)-N\_{X^\prime}(p^e)|$ has the same bound for all $p\geq p\_0$ and all $e\geq1$;


(ii) Base changing to a suitable finite extension of $\mathbf{Q}$, the value becomes constant.
3. In the special case of 2. when the bound is equal to 1, then either the difference is a constant, a quadratic character, or the negative of a quadratic character.
4. Let $B(X):=\sum\dim H^i\_c(X(\mathbf{C}),\mathbf{Q})$. Suppose $N\_X(p)\neq N\_X(p^\prime)$ for an infinite set of $p$. Then,


the set of such $p$ has density $\geq\frac{1}{(B(X)+B(X^\prime))^2}$.


Here the group theory statement is the following: let $G$ be a compact group (and set its total Haar measure to 1) and let $K$ be a locally compact field of characteristic $0$. If $\rho\_i:G\rightarrow \mathrm{GL}(n\_i,K)$ are two continuous linear representations and $a:=\mathrm{Tr}\rho\_1-\mathrm{Tr}\rho\_2$, then either


(i) $a=0$, or


(ii) {$g\in G:a(g)\neq0$} has volume $\geq (n\_1+n\_2)^{-2}$.


There's a little bit more he covered, and I've also left out the proofs. But that should give you a good idea of what his talk was about.


