Felipe Voloch complained that I didn’t list Ruby’s BBQ in my last post as one of the charms of visiting UT. I’ll make it up to him by observing that one of the charms of visiting UT is talking math with Felipe! He asked me an interesting question, about which we had different intuitions — I’ll present the question here and those readers who have an opinion are encouraged to voice it. (Math below the fold to avoid shocking the modesty of non-mathy readers.)
Let X be a curve of genus g over the field F_q. We know by the Weil conjectures (actually, the Weil theorems, since he proved them in the case of curves) that
If q is very large compared to g, one sees that
and, in particular, the number of points on X is not sensitive to the genus.
When q is fixed and g grows, the story is quite different. The Weil bounds show that
but this bound isn’t sharp: Drinfeld and Vladuts showed the better bound
Let A_q be the lim sup, over all curves X/F_q of all genera, of |X(F_q)|/g(X). The Drinfeld-Vladuts bound tells us that
and in fact this inequality is an equality when q is a square; various modular curves provide examples of curves meeting the bound. When q is not a square, I don’t think very much is known.
But this is not Felipe’s question. He asked: what can be said about the number of points on an average curve of genus g over F_q, when g is large compared to q? In other words, define
where the sum is over all isomorphism classes of curves X/F_q of genus g. In particular, is B_q equal to 0? Felipe guesses it isn’t. I guess it is. What do you think?
(Note that , the size of the set we are averaging over, is itself a pretty mysterious quantity! It’s otherwise known as the number of F_q-rational points on M_g. A preprint of de Jong and Katz, available on de Jong’s home page if you can read .dvi, gives an upper bound, which is presumably far from the truth for most g and q.)
Perhaps a relevant question is the following. When g is very large, |X(F_q)| is essentially the trace of Frob_q acting on the etale H^1 of X; we can think of Frob_q concretely as a 2gx2g matrix in the generalized symplectic group GSp_2g. More precisely, it is in the coset of Sp_2g which multiplies the alternating form by q. Denote this coset by C_q.
It is pleasant and increasingly customary to guess that Frob_q behaves like a random element of C_q. And if we let q get large with g fixed, this kind of guess can be proven correct by the Weil conjectures.
When q is fixed and g grows, the story is different. Indeed, a random element of C_q might well have trace less than -q-1; this isn’t possible for Frob_q, since X can’t have a negative number of points!
So here’s a question for random matrix lovers:
QUESTION: Let g be very large relative to q, and let M be a random element of C_q, conditional on the fact that
for all k > 0. What is the expected value of Tr(M)/g? Especially: is it 0?