## Happy birthday, Dick Gross

Just returned from Dick Gross’s 60th birthday conference, which functioned as a sort of gathering of the tribe for every number theorist who’s ever passed through Harvard, and a few more besides.  A few highlights (not to slight any other of the interesting talks):

• Curt McMullen talked about Salem numbers and the topological entropy of automorphisms of algebraic surfaces (essentially the material discussed in his 2007 Arbeitstagung writeup.)  In particular, he discussed the fact that the logarithm of Lehmer’s number — conjecturally the “simplest” algebraic integer — is in fact the smallest possible positive entropy for an automorphism of a compact complex surface.  Here’s a question that occurred to me after his talk.  If f is a Cremona transformation, i.e. a birational automorphism of P^2, then there’s a way to define the “algebraic entropy” of f, as follows:   the nth iterate of f is given by two rational functions (R_n(x,y),S_n(x,y)), you let d_n be the maximal degree of R_n and S_n, and you define the entropy to be the limit of (1/n) log d_n.  Question:  do we know how to classify the Cremona transformations with zero entropy?  The elements of PGL_3 are in here, as are the finite-order Cremona transformations (which are themselves no joke to classify, see e.g. work of Dolgachev.)  Are there others?
• Serre spoke about characters of groups taking few values, or taking the value 0 quite a lot — this comes up when you want, e.g., to be sure that two varieties have the same number of points over F_p for all but finitely many p, supposing that they have the same number of points for 99.99% of all p.  The talk included the amusing fact that a character taking only the values -1,0,1 is either constant or a quadratic character.  (But, Serre said, there are lots of characters taking only the values 0,3 — what are they, I wonder?)
• Bhargava talked about his new results with Arul Shankar on average sizes of 2-Selmer groups.  It’s quite nice — at this point, the machine, once restricted to counting orbits of groups acting on the integral points of prehomogenous vector spaces, is far more general:  it seems that the group of people around Manjul is getting a pretty good grasp on the general problem of counting orbits of bounded height of the action of G(Z) on V(Z), where G is a group over Z (even a non-reductive group!) and V is some affine space on which G acts.  With the general counting machine in place, the question is:  how to interpret these orbits?  Manjul showed a list of 70 representations to which the current version of the orbit-counting machine applies; each one, hopefully, corresponds to some interesting arithmetic enumeration problem.  It must be nice to know what your next 70 Ph.D. students are going to do…

Dick has a lot of friends — the open mike at the banquet lasted an hour and a half!  My own banquet story was from my college years at Harvard, where Dick was my first-year advisor.  One time I asked him, in innocence, whether he and Mazur had been in graduate school together.  He fixed me with a very stern look.

“Jordan,” he said, “as you can see, I am a very old man.  But I am not as old as Barry Mazur.

## Bilu-Parent update

The result of Yuri Bilu and Pierre Parent that I blogged about last summer has appeared in a new, modified version on the arXiv. The authors discovered a mistake in the earlier version — their theorem on rational points on X^split(p) is now conditional on GRH, while they get an unconditional version for points on X^split(p^2). The dependence on GRH (Proposition 5.2 in the new version) is via explicit Chebotarev bounds; under GRH one has that if E/Q is a non-CM elliptic curve whose mod-p Galois representation lands in the normalizer of a split Cartan, then p << log (N_E)^(1+eps). The idea is that when E is not CM, one can find a nonzero Fourier coefficient a_l with l at most (log N_E)^(2+eps), which is required to reduce to 0 mod p; this immediately implies the desired bound on p. In the old version, the unconditional weaker bound p << (height(j(E)))^2, due to Masser, Wustholtz, and Pellarin, was sufficient; in the present version, it’s this bound that gives you control of X^split(p^2)(Q).

## F_1, buildings, the braid group, GL_n(F_1[t,1/t])

It used to be you had to talk about “the field with one element” very quietly, and only among people whose opinion of you was secure. The reason, of course, is that there is no field with one element. Which doesn’t stop people from saying “But if there _were_ a field with one element, what would it be like?”

Nowadays all kinds of people are musing about this odd question in the bright light of day, and no one finds it kooky. John Baez covered the basics in a 2007 issue of This Week’s Finds. And as of a few weeks ago the field with one element has its own blog, “Ceci N’est Pas Un Corps.”

From a recent post on CNPUC, I learned the interesting fact that the braid group on n strands can be thought of as GL_n(F_1[t]).

So here’s a question: what is GL_n(F_1[t,1/t])?