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.*“

Diller-Favre classification of bimeromorphic maps on surfaces(Amer. J. Math. 123 (2001), no. 6, 1135–1169) is related to your question above. They show, in particular, that a Cremona transformation $f$ with zero algebraic entropy is in one of the following classes: (a) some power of $f$ in some birational model is an automorphism isotopic to the identity (the sequence of degrees is bounded); (b) $f$ preserves a pencil of rational curves (degrees grow linearly); or (c) $f$ preserves a pencil of elliptic curves (degrees grow quadratically).

As a number theorist who has passed through Harvard, I was sorry not to be able to attend last week’s conference but fascinated by your tidbits. The characters that Serre refers to, that take just values 0 and 3, have kernel where the values are 3. Factoring by that kernel gives a character that’s 0 off the identity so is a multiple of the regular representation’s character. So he’s just talking about groups with cyclic quotient of order 3 and then take the regular representation of Z/3.

I presume that Serre was talking about virtual characters. If G = SL_2(F_p) and p = 1 mod 3, then G has representations U, V, W of dimensions p, p+1, and 1 (= trivial) such that [U] – [V] + [W] only takes the values 0 and 3.

Yes, Serre was certainly talking about virtual characters (I saw him give a similar lecture in April).

The “characters” I was considering are “generalized characters”, i.e. differences of

two effective characters. And there are indeed plenty non trivial ones with

values 0,3, e.g. with G = PGL(2,p).

J-P.Serre

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