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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!

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Jordan Ellenberg has a blog post mentioning Serre's talk at Dick Gross's birthday conference here : quomodocumque.wordpress.com/2010/06/07/… It might be worth emailing him to see if he has notes. –  Andy Putman Oct 26 '10 at 4:46
    
In the second paragraph, I guess you mean "two-dimensional representation"? –  François Brunault Oct 26 '10 at 14:51
    
@Francois: Thanks, you are right, I corrected it. –  Ying Zhang Oct 26 '10 at 22:20

3 Answers 3

up vote 12 down vote accepted

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.

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Thanks! These are the things I was looking for. –  Ying Zhang Oct 26 '10 at 21:57

You might be interested in Serre's recent book

Lectures on $N_X(p)$, AK Peters, Taylor and Francis, New York, 2011, 163 pages,

a version of which is available on his website at the Collège de France.

Serre deals with the following basic question : Let $X$ be a scheme of finite type over $\mathbf{Z}$. What can you say about the number $N_X(p)$ of points of $X$ over the finite field $\mathbf{F}_p$, as $p$ varies (over the primes) ? For example, it is not entirely trivial (Theorem 1.1) to show that $X$ is empty if and only if $N_X(p)=0$ for all sufficiently large $p$.

Another aspect of the question is the relationship between the analytic space $X(\mathbf{C})$ and the sequence $N_X(p)$. For example, can you recover the dimension of $X(\mathbf{C})$, or the number of irreducible components of $X(\mathbf{C})$ from the sequence $N_X(p)$ ? See Theorem 1.2 for this.

Later (Thoerem 6.15) he discusses what conclusion can be drawn if $|N_X(p)-N_Y(p)|<2$ for a set of $p$ of density $1$.

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Last year Serre gave a talk in Moscow

http://www.mathnet.ru/php/presentation.phtml?option_lang=rus&presentid=331

entitled

``Variation with $p$ of the number of solutions mod $p$ of a family of polynomial equations"

where he discussed a "higher-dimensional" refined variant of Sato-Tate conjecture about the equidistribution of "angles" of Frobenius elements for elliptic curves without CM. Serre's conjecture relates the number of solutions with values of certain virtual characters of a certain compact Lie group (whose identity component is a maximal compact subgroup of the Mumford-Tate group involved). I believe that (in a less elementary form) this conjecture is discussed among others in Serre's Seattle talk

``Propriétés conjecturales des groupes de Galois motiviques et des représentations $\ell$-adiques"

at the Motives conference (1991). (The text has appeared in Proc. Sympos. Pure Math., 55, Part 1 published by the AMS).

Judging by the title, there might be some overlapping between Serre's talks in Zuerich/Boston this year and in Moscow last year. However, the video of Moscow talk and the notes of Seattle's talk are available.

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Thanks very much for sharing the link and the paper! I watched the video just now, it was given very nicely and clearly. For people who are interested in watching this video also, the last 30 mins is more closely related to what Rob.H explained above. –  Ying Zhang Oct 26 '10 at 22:00
    
Welcome, Yuri Zarhin ! –  Chandan Singh Dalawat Oct 27 '10 at 3:20

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