Author: Kowalski

[The post has been slightly edited to clarify some issues of identification of people, as mentioned in the first comment (24-6-09)].

The topic of this post is obviously rather far from my usual fields of expertise, but since I’ve actually used the large sieve to say something about this topic, and since I was reminded of it by the excellent talk of A. Venkatesh during the Bruggeman conference, here are a few words…

The first thing to say is that whereas the topic may seem a tad abstract and a bit far from good old analytic number theory, this is not really true. In fact, in the cases described by Venkatesh (which corresponds to — separate — joint work with F. Calegari and N. Bergeron, and concern hyperbolic arithmetic 3-manifolds), what is involved can be made extremely concrete. Indeed, to get examples, one can take an imaginary quadratic field K, an integral ideal n in K, and ask about the structure of the finitely-generated abelian group defined as the abelianization

$latex \Gamma_0(\mathbf{n})^{ab}=\Gamma_0(\mathbf{n})/[\Gamma_0(\mathbf{n}),\Gamma_0(\mathbf{n})]\subset $

of the congruence subgroup

$latex \Gamma_0(\mathbf{n})=\{\left(\begin{array}a&b\\c&d\end{array}\right)\quad,\quad \mathbf{n}\mid c\}\subset PGL(2,\mathbf{Z}_{K})$

(using the “Bordeaux” notation for the ring of integers of K; amusingly, it seems the post I once wrote on this subject has disappeared, although I could still find a link to it on Google).

The link with homology is that this abelianization is well-known (by the Hurewicz theorem) to be isomorphic to the first homology group

$latex H_1(X_{n},\mathbf{Z})=H_1(\Gamma_0(\mathbf{n}),\mathbf{Z})$

of the quotient manifold of the hyperbolic three space H₃ by the discrete group:

$latex X_{\mathbf{n}}=\Gamma_0(\mathbf{n})\backslash \mathbf{H}_3$

(speaking of notation, I noticed that Venkatesh behaved like an un-reconstructed right-winger in putting his discrete group on the right-hand side).

One outcome of the ongoing work that Venkatesh discussed (still partly conjectural, and encompassing much larger classes of examples) is the fact that the size of the torsion part of these groups tends to be “as large as possible”. As Venkatesh explained, this means roughly that this torsion group is of order exponential with respect to the norm of the ideal n, which is interpreted as being roughly (up to multiplicative constants) the volume of the manifold. The fact that this is the fastest possible type of growth follows from the interpretation of the size of the homology groups (when finite, at least) as the determinant of integral matrices which have only a bounded number of bounded non-zero entries in each row and column (as n varies). A convincing numerical example was given: the torsion subgroup for an ideal of norm about 500 (if I remember right) for

$latex K=\mathbf{Q}(\sqrt{-2}),$

was roughly of order

$latex 10^{80}.$

What I had done on this torsion topic, as explained in Section 7.6 of my almost famous book on the large sieve (and also, though with some uncorrected typos, in a preliminary short note), was to show how, in the model of random (compact connected orientable) 3-manifolds suggested by N. Dunfield and W. Thurston, something quite similar was happening with high probability.

More precisely, the 3-manifolds of Dunfield and Thurston are obtained as follows: one fixes first an integer g at least 2, then a symmetric set of generators of the mapping class group Γ_g of a closed (topological) surface of genus g, then one forms the random walk on this discrete group (where the generators are chosen uniformly and independently), getting a sequence

$latex \varphi_0=1,\quad \varphi_k=\xi_1\cdots \xi_k,$

of random variables, each of which lives in the mapping class group. Because the latter is a big complicated non-abelian group (as can be seen, for instance, quite naively from the important fact that it has a natural surjection

$latex \Gamma_g\rightarrow Sp(2g,\mathbf{Z}),$

to an integral symplectic group), this random walk is highly transient and “escapes to infinity” quickly. Now, to get from these surface maps to 3-manifolds, one uses “Heegard splittings”: given any mapping class φ, take two filled “handlebodies” of genus g (which are compact 3-manifolds with boundary, the boundary being a standard surface of genus g), then glue them together by identifying the two boundaries (which are, topologically, the same surface) with the help of the homeomorphism φ. [This procedure was invented as early as 1898 by Heegaard; personally, I have the greatest trouble visualizing it, and this has the effect that I find extremely puzzling the proofs I have seen of the fact that any compact 3-manifold can be obtained in this way, at least for some g, since it is more or less treated as close to trivial once the 3-manifold is triangulated…]

In any case, for a 3-manifold presented as a Heegaard splitting of genus g, the abelianization of the fundamental group of the resulting manifold M_φ turns out to be easy to describe in terms of the mapping class used for the identification of the handlebodies: we have

$latex H_1(M_{\varphi},\mathbf{Z})\simeq \mathbf{Z}^{2g}/\langle J,\varphi^*J\rangle,$

for some lattice J in Z^2g, where the mapping class acts from the fact that the homology of the boundary surface of genus g is a free group of rank 2g.

This formula suggests that, “generically”, the homology group of a 3-manifold should be finite, because, equally generically, the lattice J and its image under the mapping class should be transverse and generate the whole Z^2g together. This was checked qualitatively by Dunfield and Thurston, and using sieve, it was not hard to provide an exponential decay of the probability that the homology is infinite. Then, the main point, already explained by Dunfield and Thurston, is that after fixing a prime p, the analogue formula

$latex H_1(M_{\varphi},\mathbf{F}_p)\simeq \mathbf{F}_p^{2g}/\langle J\otimes\mathbf{F}_p,\varphi^*(J\otimes\mathbf{F}_p)\rangle$

holds for the homology with coefficient in the finite field with p elements, which is also equal to

$latex H_1(M_{\varphi},\mathbf{F}_p)=H_1(M_{\varphi},\mathbf{Z})\otimes \mathbf{F}_p.$

However, there is now a definite probability, of size roughly 1/p, that this homology group be non-zero (roughly, it is the probability that a determinant modulo p, which is mostly equidistributed, vanish). Because the series of 1/p over primes diverges, it follows that, intuitively, the first homology group of a random 3-manifold (in their sense at least) is typically finite, but is divisible by many primes (and hence is quite large).

Making this quantitative (as I did) is not particularly difficult once the large sieve is properly setup in discrete groups with Property (T) (or, in that case, for those having a quotient with property (T), namely the symplectic group Sp(2g,Z); Andersen has shown that the mapping class group itself does not have this property), and it leads to the conclusion that, with probability exponentially close to 1 as the length of the random walk increases, the first homology group is finite with order divisible by roughly

$latex \log k$

distinct primes. This implies (by Chebychev-type estimates) a growth which is superpolynomial but potentially less than exponential, namely the size should be at least

$latex k^{\alpha\log\log k}$

for some constant α>0, with probability close to 1. If one could show that the primes dividing the order are not always too small (a property which, by the way, was clearly visible in the example shown by Venkatesh, with some fairly large primes appearing), it would follow that the size should be exponential in k, since this is the typical size of an integer with that many prime factors.

At this point one may object that this parameter k, the length of the random walk, is not a good parameter to measure the 3-manifold M_φk with, because it not canonical at all. In fact, although all (compact connected orientable) 3-manifolds have a Heegaard splitting, its genus g can not be chosen at will (one can not always take g=2, as far as I understand, though apparently this is possible for many 3-manifolds, this being related with the minimal number of generators of the fundamental group). Moreover, for a given genus, there are many choices of generating sets to define the random walk, and even then the random walk might come back a few times to the same 3-manifold with different values of k, etc… This suggested that only the asymptotic behavior (as the length of the walk grows) of things like the average number of coverings of the manifold with a given Galois group, etc, really carried significance (such numbers were computed precisely in the paper of N. Dunfield and W. Thurston).

The talk of Venkatesh reminded me of one possible rough interpretation of k, which I vaguely remembered from the paper of Dunfield and Thurston: it is the content of what they state as Conjecture 2.11 (page 12), which says that, with high probability, the random 3-manifold should be hyperbolic, and that its expected volume should grow linearly with k. And if we interpret k as a succédané of the volume, then an exponential lower bound corresponds exactly to the growth conjectured by Bergeron and Venkatesh, while the almost exponential one obtained from the sieve gives strong evidence for it being, indeed, “generic”. [Their manifolds being of a special type, the generic case does not apply directly, of course.]

Here I would have been happy to give some convincing comments on the status of this conjecture of Dunfield and Thurston. J. Maher has a preprint where he proves the hyperbolicity statement, but the proof depends on the geometrization conjecture of Thurston which was, of course, proved by (the methods of) Perelman. Maher also states that the volume part follows then from his result and a work of Brock and Souto, but a web search only reveals this paper to be “in preparation”; added to the fact that I can not claim to understand even Maher’s proof, I feel more comfortable simply saying that it seems that the statement may well be on the verge of being proved. [Any more informed comments will be extremely welcome!]

I would like to conclude by saying that although this is not my area, I found it to be fairly accessible, at an intuitive level, by virtue of the existence of very good surveys and expositions online, e.g., this book-in-progress of Farb and Margalit, and the paper of Dunfield-Thurston, which is very readable.

As I already mentioned, I was last week at the conference organized to honor R. Bruggeman’s 65th birthday — though both organizers and honoree claimed this was a pretext to have a conference on the analytic theory of automorphic forms in Holland.

It was a very enjoyable week, and an excellent occasion to learn more about some of the newer brands of automorphic objects which have become popular, but which I don’t really understand yet (I must admit I couldn’t tell the difference between a weak harmonic Maass form, a mock theta function, a mixed modular form, or a weak harmonic Jacobi form…). Since I used a beamer presentation for my talk on families of L-functions and cusp forms, here is a link to it. Although it is almost completely “philosophy”, and does not discuss any new result, some of the points, it may be of some interest (at some point, I will probably write a more complete post on the questions which are raised there).

The workshop was held in a conference center located about 10 minutes from Utrecht. This was a very nice location, and since everyone (except for local people) stayed in the same place, and the coffee was as plentiful as I’ve ever seen (and free), the evenings were quite social affairs. During the Thursday evening barbecue, the question arose (at least at my table) of determining which was the silliest conjecture that people had seriously spent time on. Various glorious names were suggested by participants: Goldbach, Fermat, ABC, etc (I will of course hide the identity of those who made these propositions…) A semi-popular favorite was the Lehmer conjecture which claims that

$latex \tau(n)\not=0,\quad\text{ for } n\geq 1,$

i.e., the non-vanishing of the Ramanujan tau-function. Recall that the latter is defined by the formal power-series expansion

$latex \sum_{n\neq 1}{\tau(n)q^n}=q\prod_{n\geq 1}{(1-q^n)^{24}},$

and that Ramanujan had conjectured some of its remarkable properties, including the multiplicativity

$latex \tau(mn)=\tau(m)\tau(n)\quad\text{ if } m \text{ and } n \text{ are coprime},$

and the bound

$latex |\tau(n)|\leq n^{11/2}d(n)$

where d(n) is the number of positive divisors of n. The latter (which is generalized to the “Ramanujan-Petersson conjecture”) was proved by Deligne as a very deep consequence of the Riemann Hypothesis over finite fields.

As a matter of fact, this conjecture of Lehmer had been the topic of one of the morning lectures; E. Bannai had explained his work with T. Miezaki, which gives an interpretation of the Lehmer conjecture in terms of properties of the E₈ lattice and spherical design properties of its shells. This suggests that the conjecture is more than a random guess that has every chance to be true, but for no good reason.

In the end, I believe a consensus arose that, at least, the following strengthening of the Lehmer conjecture is an extremely silly question:

“Conjecture”. The tau function is injective.

(this is stronger than Lehmer’s conjecture because if some τ(n) is zero, then by multiplicativity, many others will also be).

This is a question which I had raised (to myself) after reading a paper of Garaev, Garcia and Konyagin which shows, using quite clever arguments, that the Ramanujan function takes “many” different values; at the time I checked it was valid for the largest table of tau that I could find by a quick googling. If there are bigger ones now easily available and obvious counterexamples, I will of course emphasize that this was just a random guess that had every chance to be true for no good reason.

(Note that the example of Hardy apologizing for discussing the tau-function, as seemingly part of the “backwaters of mathematics”, means one must be careful with judgments of value about mathematical problems based on one’s current understanding…)

I was in Paris part of this week-end, where I had planned to attend the Bourbaki Seminar lecture of E. Breuillard on the recent works of Einsiedler, Lindenstraus, Michel and Venkatesh. Embarrassing scheduling mistakes on my part forced me to miss it, however, and to take an earlier train to Holland (where I am now, attending the conference on the analytic theory of automorphic forms coinciding with the 65th birthday of R. Bruggeman). Fortunately, I did manage on Saturday to pick up a copy of the physical Bourbaki report, so this was not entirely a disaster.

From Paris to Holland, I took the Thalys train, which is distinguished by having wireless on board and by quadrilingual announcements: French, because it starts from Paris and goes through parts of French-speaking Belgium, Dutch, because it crosses also Dutch-speaking Belgium and one branch reaches Holland, German, because of the other branch going to Köln, and finally English, for good measure.

In this multilingual environment, one notices that the names of a number of towns changes with the language; for instance, Köln is Cologne (famous for its eau) in French and English. The prize of variation on this trip was Liège, alias Luijk, alias Lüttich (alias Liege, if one wants to be picky). Hence the question of the day: which town has the most different names? (Say within same-alphabet countries, to avoid issues of transliteration).

This week, the Forschungsinstitut für Mathematik is host to a conference to honor the 61st birthday of G. Wüstholz — of course, diophantine approximation, arithmetic geometry, and related areas were very much in focus. (I write “is” because, although this is most definitely Friday evening, there are still two talks scheduled tomorrow morning).

The programme was very enticing, so I attended most of the talks, despite not knowing much about some of the topics (e.g., Arakelov geometry). Among those I found especially interesting (partly because I was at least a bit more au courant) were the following:

(1) U. Zannier explained some recent work with D. Masser which can be described as trying to understand (and devise methods to study) intersections of “sparse” sets of arithmetic interest. The concrete example he described, which had been the original question of Masser, was the following: consider the Legendre family of elliptic curves

$latex E_t : y^2=x(x-1)(x-t)$

and the points

$latex P_t=(2,p_t),\quad Q_t=(3,q_t)$

on E_t (so there are two choices of the y-coordinates, which will not affect the question). What can one say about the set of parameters t for which both

$latex P_t,\text{ and } Q_t$

are torsion points on the curve E_t? It is not difficult to check that for either of the two points, there are infinitely many such parameters, forming “sparse” sets, and the results (or rather, the methods) of Masser and Zannier imply, in particular, that these two have at most a finite number of intersection points, i.e., that there are only finitely many t for which both are torsion points.

One may wonder why the question should be of any interest (I personally find it very nice), but Zannier emphasized that the new techniques they had to devise were quite significant and very likely to be useful in many contexts. These techniques are quite novel in this area, and rely ultimately (and quite strikingly) on the circle of ideas that started with the 1989 work of Bombieri and Pila on the number of rational (or integral) points on transcendental curves (in the plane, say). Zannier illustrated the link with a sketch of a new proof of the original Manin-Mumford conjecture (first proved by Raynaud; it states that if an algebraic curve defined over a number field is embedded in its Jacobian variety, then the curve only contains finitely many torsion points): using a transcendental parametrization, we can see the curve as a transcendental curve

$latex C\subset J(C)=\mathbf{C}^g/\Lambda$

for some lattice

$latex \Lambda\subset \mathbf{C}^g,$

whereas the torsion points are the elements of

$latex \Lambda\otimes \mathbf{Q}/\Lambda$

and thus have rational coordinates. The intersection is thus, in principle, similar to the situation of Bombieri-Pila. Of course, much more work is required, and Zannier said that the extension to deal with Masser’s question require rather more subtle versions of the Bombieri-PIla ideas, including the very recent ones of Pila-Wilkie where — to add more fun to the mix — logic enters the game through the consideration of transcendental varieties with graphs definable in an o-minimal structure. (About which, despite looking at the book of van den Dries, I am still terribly ignorant).

See here for the Compte Rendus note announcing this result; Zannier said the full paper is almost ready, and one can see more applications of this type of methods in this recent paper of Pila.

(2) T. Shioda gave a very nice lecture — full of beautiful examples — on recent work of his concerning the determination and structure of the (finite) set of integral sections of an elliptic surface over the projective line (see his preprint); for rational elliptic surfaces, he explained a very beautiful general description involving commutative algebra. In particular, there are then at most 240 integral sections. For instance, there are exactly 240 polynomials

$latex (p(t),q(t))\in \mathbf{C}[t]\times \mathbf{C}[t]$

such that they are points on the rational elliptic surface with equation

$latex y^2=x^3+t^5+1;$

indeed, those points, for the height pairing, can be identified with the 240 vectors of minimal length (squared) 2 on the famous E₈ lattice.

(3) Y. Bilu explained his recently recovered-from-the-edge proof, with P. Parent, of the “split Cartan” case of the Serre uniformity question concerning the maximality of the Galois action on torsion points of prime order on elliptic curves over the rationals (see also this post for more background information– though as indicated, the first proof had a mistake, which was corrected in February–March this year, the overall strategy has remained the same). The preprint of Bilu and Parent is on arXiv.