Question

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. Suppose

   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.