## Are ranks bounded?

Important update, 23 Jul:  I missed one very important thing about Bjorn’s talk:  it was about joint work with a bunch of other people, including one of my own former Ph.D. students, whom I left out of the original post!  Serious apologies.  I have modified the post to include everyone and make it clear that Bjorn was talking about a multiperson project.  There are also some inaccuracies in my second-hand description of the mathematics, which I will probably deal with by writing a new post later rather than fixing this one.

I was only able to get to two days of the arithmetic statistics workshop in Montreal, but it was really enjoyable!  And a pleasure to see that so many strong students are interested in working on this family of problems.

I arrived to late to hear Bjorn Poonen’s talk, where he made kind of a splash talking about joint work by Derek Garton, Jennifer Park, John Voight, Melanie Matchett Wood, and himself, offering some heuristic evidence that the Mordell-Weil ranks of elliptic curves over Q are bounded above.  I remember Andrew Granville suggesting eight or nine years ago that this might be the case.  At the time, it was an idea so far from conventional wisdom that it came across as a bit cheeky!  (Or maybe that’s just because Andrew often comes across as a bit cheeky…)

Why did we think there were elliptic curves of arbitrarily large rank over Q?  I suppose because we knew of no reason there shouldn’t be.  Is that a good reason?  It might be instructive to compare with the question of bounds for rational points on genus 2 curves.  We know by Faltings that |X(Q)| is finite for any genus 2 curve X, just as we know by Mordell-Weil that the rank of E(Q) is finite for any elliptic curve E.  But is there some absolute upper bound for |X(Q)|?  When I was in grad school, Lucia Caporaso, Joe Harris, and Barry Mazur proved a remarkable theorem:  that if Lang’s conjecture were true, there was some constant B such that |X(Q)| was at most B for every genus 2 curve X.  (And the same for any value of 2…)

Did this make people feel like |X(Q)| was uniformly bounded?  No!  That was considered ridiculous!  The Caporaso-Harris-Mazur theorem was thought of as evidence against Lang’s conjecture.  The three authors went around Harvard telling all the grad students about the theorem, saying — you guys are smart, go construct sequences of genus 2 curves with growing numbers of points, and boom, you’ve disproved Lang’s conjecture!

But none of us could.

## Positive motivic measures are counting measures

A new, very short paper with Michael Larsen, “Positive motivic measures are counting measures” is up on the arXiv today.  I thought I’d say a bit here about where the problem came from, since we don’t do so in the paper.

In the project with Akshay that I talked about at the recent Columbia conference on rational curves on varieties, one thing you do is compute estimates for |M_n(F_q)|, where M_n is the moduli space of algebraic maps of degree n from P^1 to some fixed target variety X, and F_q is a finite field.  These inequalities turn out to be very nicely uniform in q, which leads one naturally to ask; do the proofs actually give “motivic estimates” for the class [M_n] in the Grothendieck ring K_0(Var_K), for various non-finite fields K?

Well, what does it mean for one element r of a ring R to “estimate” another element s?  It might mean that r-s is rather deep in some natural filtration on R.  Those don’t seem to be the kind of estimates our methods provide; rather, they say something more like

(r-s)^2 <= B

where B is some fixed element of K_0(Var_K).  But what does “<=” mean?  Well, it means that B – (r-s)^2 is nonnegative.  And what does “nonnegative” mean?  That’s the question.  What the proof really gives is that B – (r-s)^2 lies in a certain semiring N of “nonnegative motives” in K_0(Var_K).  Let’s not be too precise about what N is; let’s just say that it includes [V] for every variety V, and it has the property that |n(F_q)| >=0 for all q, whenever n lies in N.  In particular, that means that

(|r(F_q)| – |s(F_q)|)^2 <= B(F_q)

so that, on the level of counting points, s(F_q) really is a good estimate for r(F_q).

So one might ask:  are there other interesting positive motivic measures — that is, homomorphisms

f: K_0(Var_K) -> reals

which take N to nonnegative reals?  If so, f(s) would be a good estimate for f(r).

And the point of this note with Larsen is to say, with some regret, no — any motivic measure which assigns nonnegative values to the classes of varieties is in fact just counting points over some finite field.  Which sort of kills in its crib the initial hope of some exciting world of “motivic inequalities.”

Of course, the reals are not the only ordered ring!  As Bjorn Poonen pointed out to me, for a general field K you can find an ordered ring A and a measure

g: K_0(Var_K) -> A

which is positive in the sense that g sends every variety to an element of A greater than or equal to 0; these come from big ultraproducts of counting measures of different finite fields.  Whether these measures are “interesting” I’m not sure.