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 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!