E. Kowalski’s blog

From the CERN English website:

CERN’s Visits Service organises tours of its experimental areas and facilities, which are free of charge. Tours in several languages are organised on Mondays to Saturdays starting at 9 a.m. and 2 p.m. It is essential to book in advance.
Please note that the tours are not suitable for children under 14 years of age.

(emphasis mine, as people say in history books).

Now from the French version:

Le service des visites du CERN organise des visites gratuites de sites d’expériences et d’infrastructures. Ces visites sont proposées en plusieurs langues, du lundi au samedi, à 9h ou à 14h. La réservation est obligatoire.
Veuillez noter que le niveau des visites n’est pas adapté aux enfants de moins de 10 ans.

I’m still thinking aloud (or the bloggerly equivalent thereof) about the topic of my last post, and I’m at this delightful stage of guessing there may well be interesting questions there, and yet not knowing too precisely which ones are easy, which are impossible, or even which are already hidden in the maze of MathSciNet under cleverly disguised search terms.

So consider the case of G=SL(2,Z) again, and assume given a subgroup H. In broadest terms, we’re trying to identify which conjugacy classes in G have representatives in H. We can’t exclude that all of them do; if that happens, we know that (1) H is of infinite index (see the first comment by D. Speyer to the earlier post); (2) but H surjects, by reduction modulo p, to SL(2,F_p) for every p. The latter condition implies in particular that H be Zariski-dense in SL(2) (otherwise, its reduction would be in G(F_p) for some proper algebraic subgroup, and this would be strictly contained in SL(2,F_p) if p is large enough). Nicely enough, such subgroups (especially when finitely generated) are currently the topic of much work in terms of spectral theory, expansion and the like (see for instance these recent preprints by Bourgain, Gamburd and Sarnak, and by Bourgain and Kantorovich).

The conjugacy classes of G have been classified for a long time (for instance, this is needed for the Selberg Trace Formula). The most interesting, or at least those I’m going to look at first, are the so-called hyperbolic ones, which are characterized by the fact that, for some (unique) a>1, they contain a representative which is conjugate in SL(2,R) to

which acts as a dilation

on the Poincaré upper half-plane. A more direct characterization, in terms of an arbitrary representative g of the conjugacy class, is that

So, for instance, we can take the conjugacy class of

In the case of a conjugacy class in G, the dilation a is a real quadratic integer (it is the largest eigenvalue of the matrix, and the determinant, which gives the constant term of the minimal polynomial, is 1). In the example above, we get

In SL(2,R), the dilation is the unique invariant of a hyperbolic conjugacy class (and visibly any a>1 occurs as a dilation). In G, things get a bit more arithmetic (which means more complicated, though the two words are maybe not quite synonyms). Essentially (I am here forgetting or glossing over some important semi-technical issues), for a given discriminant

there are only finitely many G-conjugacy classes, and the number of them is the class number of the associated real quadratic field. (Precise details are given in this old paper of Sarnak).

From my point of view of conjugacy classes, the following seems the obvious salient features:

(1) to have a chance to find a given hyperbolic conjugacy class in a subgroup H, a necessary condition is that H contains a matrix with a certain trace (up to sign; if we assume that minus the identity is in H, the sign ambiguity disappears); this condition, in turn, is obviously susceptible to local congruence obstructions — but we know that for a Zariski-dense (finitely generated) subgroup of G, all but finitely many of these congruence obstructions modulo primes will vanish by Strong Approximation.

(2) if we have a subgroup where all local obstructions disappear (for instance, all reductions modulo primes are surjective; not I don’t actually have an example of a proper subgroup of infinite index where this holds…), we are led to wonder whether all ideal classes associated with hyperbolic elements of G have representatives in H; this question is reminiscent of the representation problem for integers by ternary definite quadratic forms (where there are fairly simple necessary conditions for this to happen, and those are fairly classically also sufficient for an integer to be representation by some form in the same genus as the given one, which means by some form everywhere locally equivalent to it, while the representability by the given form holds for sufficiently large integers by much deeper work involving Fourier coefficients of half-integral modular forms — a very beautiful story, where crucial work is due to Iwaniec and Duke and Schulze-Pillot).

As before, hopefully more to come…

A fairly well-known fact about finite groups says that if H is a subgroup of G, and H intersects every conjugacy class in G, then in fact H=G. This is quite useful, for instance, for some problems of Galois theory, because one might have to understand a finite group using information only about which conjugacy classes it represents in a bigger group (e.g., a Galois group represented as permutation groups of the roots of an integral polynomial, where the factorization of the polynomial modulo various primes indicates which conjugacy classes of the corresponding symmetric group intersect the Galois group; I’ve already mentioned this type of things here and here).

It is natural to ask what happens with other kinds of groups. The example of compact Lie groups shows that if G is infinite, there may well exist a subgroup H intersecting every conjugacy class; for instance, if G=U(n), every element can be diagonalized, i.e., every element is conjugate to one in the subgroup H of diagonal matrices (which, if n is not 1, is not the same as G…) However, these are quite special groups, and one might suspect that some interesting infinite groups retain this property (which I’ll call the Jordan property here, as suggested by Serre’s nice paper about this theorem of Jordan).

Although I’ve started looking around, I haven’t found much information yet on this. The first groups I’m trying to understand are arithmetic groups like G=SL(n,Z). Here’s one simple example in such a case: if n is at least 3, then G has the Jordan property “with respect to finite index subgroups” (i.e., any finite index subgroup intersecting all conjugacy classes of G is equal to G). This requires a fairly big hammer, but is otherwise very easy: by the Congruence Subgroup Property, any H of finite index satisfies

for some integer d, where

is a principal congruence subgroup. This means that, for some subgroup Γ of the finite quotient G/G(d), we have

but then it is immediate (by lifting to H) that if H intersects all conjugacy classes of G, then also Γ intersects every conjugacy class in G/G(d), and we get

from the finite group case, and therefore H=G.

More generally, one sees at least (without using the Congruence Subgroup Property) that if H is a subgroup of G=SL(n,Z) intersecting every conjugacy class, then we have

for all d (because the reduction modulo d maps G surjectively to SL(n,Z/dZ) for all d). However, this condition is not as stringent as it may look: it is known (the “Strong Approximation Theorem” of Mathews, Vaserstein and Weisfeiler) that this holds, at least for all integers d coprime with some “conductor” f, for any subgroup of SL(n,Z) which is Zariski-dense in SL(n), and such groups can be quite “small”. However, one might intuitively hope that, being “smaller” than finite index subgroups, they would intersect fewer conjugacy classes (?). On the other hand, I also don’t know offhand of a non-trivial subgroup with conductor f=1…

For the special case n=2, the Congruence Subgroup Property fails (one way to see it, as explained in this survey of Raghunathan, is to contrast the fact that SL(2,Z) has finite quotients like the alternating group A₅, whereas any non-abelian simple quotient of a congruence subgroup is of the type SL(2,Z/pZ) for a prime p, and none of these is isomorphic to A₅, simply because none is of order 60). Then it’s not clear to me if some finite index subgroup (not of congruence type) could intersect every conjugacy class of SL(2,Z).

Hopefully, I’ll have the occasion to write more about this as I explore the literature…

Nature, stretching Horace Davenport out, had forgotten to stretch him sideways, and one could have pictured Euclid, had they met, nudging a friend and saying: “Don’t look now, but this chap coming along illustrates exactly what I was telling you about a straight line having length without breadth.”

(taken from the first pages of Uncle Fred in the Springtime; due to a rather unfortunate fall in the stairs last week, I have to rest and watch my back for a few days, and hence I’ve been in need of light and refreshing literature to pass the time, turning therefore in part to re-reading some novels of P.G. Wodehouse.)

Every time I have found a mistake in one of my books (or paper), I have had to repeat two or three times the well-known mantra The only way to not make mistakes in print is to not publish anything, before I could regain some composure. If the mistake was indicated by a reader, it is even worse. Of course, the problem often turns out to be mostly innocuous — e.g., the definition of “equidistribution” in my book with Henryk Iwaniec is wrong (we forgot to restrict the open sets for which convergence is claimed to those where the boundary has limiting measure zero), but it’s hard to imagine much harm coming from this. However, more serious errors are more disturbing; in particular, it makes you wonder what the frequent casual claim that The responsibility for all mistakes of course lies with [the author] really means in practice, when honest readers may well be led to a lot of misguided effort because of what is claimed with characteristic authorial confidence.

To give an unfortunate example, B. Conrey just pointed out that the same book has a very unfortunate claim that the local factor of the Rankin-Selberg convolution of two classical (holomorphic or Maass) modular forms is “the obvious one” for a ramified prime p which appears with exponent exactly one in the l.c.m. of the conductors of the two factors. In particular, this means that we claim that if f (resp. g) has p-factor

corresponding (in the case of elliptic curves) to split/non-split multiplicative reduction at p, then the local factor for the Rankin-Selberg convolution is

which is in particular of degree 1 in p^-s. Alas, alas, this is quite wrong; the local factor should be of degree 2! The simplest way to see this is probably to think in terms of the local Langlands correspondence (note that one doesn’t need to know it is a theorem to apply it heuristically): the local factors for each form are supposed to be of the type

where ρ is a 2-dimensional representation of some local Galois group acting on V, F_p is the Frobenius at p, and I is the inertia group. Generically one might indeed assume that if ρ₁ and ρ₂ have each a one-dimensional invariant subspace, the tensor product (which corresponds to the Rankin-Selberg convolution) would also have the same property (with basis

of the invariant subspace, where e corresponds to that of ρ₁ and f to that of ρ₂. But the classification of these things (which are among the so-called Steinberg representations) shows that it is possible (it might indeed be always the case: I need to brush up my understanding of all this before making claims here!) that

is a twist of ρ₁ by a character of degree 1. Then this means that one can find a basis e, f of a common underlying two-dimensional space so that I acts by

and then, of course, we see that both vectors

in the tensor product are invariant under I.

As I said, this mistake is quite annoying. My guess is that it may not have created any trouble (yet) for our readers: I’m pretty sure that the claim we make is true if the prime p is ramified only for one of the two modular forms (I’ll have to find a proper reference, of course), and I don’t think many analytic applications would have been outside this case. However, I plan to look at least quickly through the list of papers on MathsciNet which refer to our book to detect possibly problematic cases…