Mathematics – Page 22 – E. Kowalski's blog

Selberg archive

Looking around the IAS website yesterday, I noticed that a selection of papers from Selberg’s archive is now available online, to be supplemented by a more comprehensive website. Mostly, these seem to be his transparencies and other notes for lectures he gave (eg., this one), rather than drafts of unpublished work. One interesting item is a transcript of an unpublished interview from 1989. Here is a quote I like (beginning after a discussion of the Chowla–Selberg formula and collaboration in general):

Yes, this was something, of course, quite different from what started it. It’s rather typical in many ways that in mathematics very often what you end up with has very little to do with what you start out with. You may start out trying to do something, and as you get into it and learn something either your attention may switch completely, — because you understand something more of the problem, perhaps what you had initially as a goal is quite impossible — or you may come across something as you are going along, quite by accident, that completely throws your attention in a completely different direction.

One can never, I think, predict where one is going when one starts out.

(p. 29 in the the first part of the interview)

Some things I also learnt or realized: Mordell was American (and not English); Selberg got the Fields medal well before he proved the trace formula (1950 against the first results in 1951–1953); Veblen used to organize wood-cutting expeditions in the IAS woods, and apparently Pauli was considered a dangerous participant with axe and saw (page 6 of the second part).

Various updates

Travel and vacation have delayed the writing of the next installment of my series of posts on “Algebraic twists of modular forms”, but I am working on it and hope to have it ready soon… In the meantime, here are two updates on some of my favorite topics from yesterposts.

First, for the bad news: if your proof of the Riemann Hypothesis depended on my earlier claim that the spectral gap for the Lubotzky group modulo primes is at least $2^{-38}$ , then you’re in trouble. This bound depended on an induction which turns out to be so mistaken that I have promised myself never more to be upset when I see a completely wrong induction proof in a student’s paper. This was found by the referee of my paper on explicit growth and expansion, who also did the most amazing job in checking all the myriad details inherent in a paper of this type. I have put up a corrected version on the web, where the spectral gap becomes again $2^{-2^{45}}$ or so. (I have also started re-reading and updating my notes on expander graphs, correcting the same mistakes, but also replacing the fully explicit version with a more readable “just show the spectral gap exists”-writeup.)

Now, for the better news: in the last few weeks, I also prepared a written version of the mini-course on “Sieve in discrete groups” that I gave at MSRI in February, on the occasion of the “Hot Topics” Workshop held there concerning “Thin groups and super-strong-approximation” (a Proceedings volume is in preparation; another excellent survey already available is the one by Rapinchuk on Strong Approximation). This was just intended to be a straightforward survey, but while finishing it, I wondered again about a question I had vaguely thought about earlier without conclusion: “Is there an Erdös-Kac Theorem in the context of the ‘affine’ sieve?” More precisely, as in the Bourgain-Gamburd-Sarnak context, let $\Gamma\subset \mathrm{SL}_r(\mathbf{Z})$ be a finitely generated subgroup, and assume (for simplicity) that its Zariski closure is $\mathrm{SL}_r$ (which means that, for all primes $p$ large enough, the reduction modulo $p$ from $\Gamma$ surjects to $\mathrm{SL}_r(\mathbf{F}_p)$ ). Let $f$ be a non-constant, integral-valued, polynomial function on $\mathrm{SL}_r(\mathbf{Z})$ . Can one prove that the number of prime factors of $f(g)$ , for a “random” element $g\in\Gamma$ , is approximately gaussian when suitably normalized?

The answer, it turns out, is “Yes”. In fact, as soon as I thought about it a bit seriously for a few minutes, I remembered a very nice paper of Granville and Soundararajan which gives a short and easy proof of the classical Erdös-Kac Theorem, and goes on to explain how their method can be generalized to study the number of prime factors in much more general sequences than the integers. It was then almost immediately clear that one can use this method to get a form of Erdös-Kac Theorem for discrete groups (and I wouldn’t be surprised if this had been noticed earlier by others.) One interesting point is that this seems to be a case where defining “random” elements of $\Gamma$ seems most natural in terms of random walks, instead of looking at balls with respect to some metric or other. In the situation above, if $S=S^{-1}$ is a generating set of $\Gamma$ , we denote by $(\gamma_n)$ a random walk on $\Gamma$ , starting at $1$ with steps taken uniformly and independently at random from $S$ . Then one shows that there exists some $\kappa>0$ such that the random variables
$\frac{\omega(f(\gamma_n))- \kappa\log n}{\sqrt{\kappa \log n}}$
converge to the standard gaussian as $n\rightarrow +\infty$ , where $\omega(n)$ is the number of primes dividing $n$ , with the convention that it is $0$ for $n=0$ .

This applies in other contexts involving sieve in discrete groups. For instance, if $(M_n)$ is a sequence of random Dunfield-Thurston 3-manifolds, and if we denote by $\omega(M_n)$ the number of primes $p$ such that $H_1(M_n,\mathbf{F}_p)\not=0$ , with the convention that $\omega(M_n)=0$ if $M_n$ has non-zero first rational Betti number, then the sequence
$\frac{\omega(M_n)-\log n}{\sqrt{\log n}}$
also converges in law to the standard gaussian.

Of course, I can’t help wondering if there could exist, as in the classical case, a finer statement of mod-Poisson convergence concerning the limit behavior of
$\mathbf{E}(e^{it \omega(f(\gamma_n))})$
for $t\in\mathbf{R}$ . This seems very hard (taking $t=\pi$ gives the average of the Liouville function) and rather mysterious… The renormalized Erdös-Kac Theorem is really about the distribution of “small” prime factors (on logarithmic scale) of integers, as one can see easily by noting that an integer $n$ has a bounded number of prime factors $p>n^{\delta}$ for any fixed $\delta>0$ ; since the theorem implies in particular that most integers have about $\log\log n$ prime factors, we see that the limiting distribution arises from integers without prime factors of such size. The mod-Poisson convergence, on the other hand, does take these factors into account, but we have currently no idea whatsoever about the distribution of “large” prime divisors of $f(g)$ in the affine sieve context…

Algebraic twists of modular forms, II

I continue here the discussion, begun in my previous posts, of my recent work with Fouvry and Michel. So, recall that we want to estimate a sum of the type

$S(f,K)=\sum_{n\geq 1}{\rho_f(n)K(n)V(n/p)}$

were $\rho_f(n)$ denotes the Fourier coefficients of a fixed modular form $f$ , $K(n)$ is some complex-valued function defined modulo a prime $p$ and extended to all integers by $p$ -periodicity, and $V$ is a test function, compactly supported in $]0,1[$

I will present the strategy rather roughly first, and then refine it. For the moment, nothing will be assumed of $K(n)$ .

[Analytic step]. Using (rather sophisticated) techniques from the analytic theory of modular forms (in particular, amplification and the Kuznetsov formula), we show that one can bound from above the sum $S(f,K)$ using some auxiliary sums. Precisely, we first need the discrete Fourier transform of $K$ , which we normalize by putting $\hat{K}(t)=\frac{1}{\sqrt{p}}\sum_{x\in\mathbf{F}_p}{K(x)e\Bigl(\frac{tx}{p}\Bigr)}$
for $t$ modulo $p$ (a unitary normalization). Then, for $\gamma\in \mathrm{PGL}_2(\mathbf{F}_p)$ , we let
$C(K;\gamma)=\sum_{z\in\mathbf{F}_p}{\overline{\hat{K}(\gamma\cdot z)}\hat{K}(z)}.$
We call these sums correlation sums. Using an auxiliary parameter $L\geq 1$ , their relation to our problem can be first expressed by
$|S(f,K)|^2 \ll p^2L^{-1}+pL\times \max_{\gamma\in \mathrm{PGL}_2(\mathbf{F}_p)} |C(K;\gamma)|$
(up to terms like $p^{\varepsilon}$ and logarithms). In fact, we do not write this down, because one can see that it useless as is: for $\gamma=1$ , the correlation sum $C(K;1)$ is typically as large as $p$ , so the right-hand side is at least of size $p^2$ , which is essentially the trivial bound (that one can get from Rankin-Selberg theory, for $K$ bounded)…
[Pruning]. This first objection to the method is however not significant. Indeed, it is clear from the proof that not all correlations sums $C(K;\gamma)$ play a role when estimating $|S(f,K)|^2$ . In particular, provided $L$ is not too large (say $L<p^{1/2}$ ), it is essentially immediate that one can replace the bound above with
$|S(f,K)|^2 \ll p^2L^{-1}+pL\times \max_{\gamma\notin B_p} |C(K;\gamma)|$
where $B_p$ is the stabilizer of $\infty$ in the action of $G_p=\mathrm{PGL}_2(\mathbf{F}_p)$ on the projective line $\mathbf{P}^1(\mathbf{F}_p)$ , i.e.
$B_p=\Bigl\{\begin{pmatrix}a& b\\\ 0&d\end{pmatrix}\Bigr\}\subset G_p.$
This means that we do not mind if some correlation sums are big, provided these “bad” sums are of specific shape, here, upper-triangular.
[Estimating]. Suppose we want to use the previous bound heuristically first for a weight $K(n)$ which is generic (or random), and bounded by $1$ . Then we can expect that the unitary Fourier transform will be basically bounded (by the philosophy of square root cancellation — for which, by the way, I would have expected that there existed a better informal introduction on the internet than in this four-year-old post), and will also be rather random. But then the correlation sum is also an oscillating sum of length $p$ , and optimistically, should be of size $\sqrt{p}$ for $\gamma\not=1$ . We can therefore hope to get
$|S(f,K)|^2\ll p^2L^{-1}+p^{3/2}L.$
The optimal choice of $L$ is $L=p^{1/4}$ , and this gives
$|S(f,K)|^2 \ll p^{2-1/4},\quad\quad\text{ or } \quad\quad S(f,K)\ll p^{1-1/8}.$

This is the limit of the method.

But can we really apply this approach to concrete functions $K(n)$ ? The reader will see that the quality of the bound here is exactly (up to epsilons) what I stated in the last post for cases like

$K_1(n)=\chi(n),\quad\quad K_2(n)=e\left(\frac{\bar{n}}{p}\right).$

However, these two cases are in fact rather different. In the second case, spelling out the definition of $C(K_2,\gamma)$ , it is easy to see that the correlations sums are, up to a factor $1/p$ , additive exponential sums in three variables (when $\gamma$ is anti-diagonal, it is a Kloosterman sum in three variables). For these, we show that there exists $M\geq 1$ such that

$|C(K_2,\gamma)|\leq Mp^{1/2}$

for all $\gamma\not=1$ , so that the argument above applies (this is a case of the Riemann Hypothesis over finite fields with optimal cancellation for sums in three variables; when $\gamma$ is restricted to anti-diagonal matrices, the fundamental result of Deligne on hyper-Kloosterman sums shows that one can take $M=4$ ).

But for $K_1(n)$ , things are different. Here it is easy to analyze the correlation sums because the Fourier transform of $\chi(n)$ , for a non-trivial character modulo $p$ , is a multiple of $\bar{\chi}(t)$ . Hence $C(K_1,\gamma)$ is a one-variable multiplicative character sum, and can be analyzed using Weil’s methods. One finds that there exists an absolute constant $M\geq 1$ such that

$|C(K_1,\gamma)|\leq Mp^{1/2}$

except if either $\chi$ is non-real, and $\gamma$ is diagonal, or if $\chi$ is real, and $\gamma$ is either diagonal or anti-diagonal (and these are genuine exceptions, if $M$ is supposed to be an absolute constant).

The first case lies within the realm of the previous discussion, but when $\chi$ is a real character, this is not true anymore.

Similarly, if

$K_3(n)=\frac{S(n,n;p)^2}{p}-1,$

as at the end of the previous post, one finds that the Fourier transform $\hat{K}_3(t)$ is exactly the coefficient $a_p(t)$ such that

$|E_t(\mathbf{F}_p)|=p+1-a_p(t),$

where $E_t$ is the elliptic curve which was discussed in the first post in this series. The relations between $|E_t(\mathbf{F}_p)|$ and $|E_{\gamma\cdot t}(\mathbf{F}_p)|$ mentioned there imply that the correlation sums $C(K_3,\gamma)$ are of size $p$ for $\gamma$ in a certain dihedral group $D$ of order $8$ , namely the setwise stabilizer of $\{-4,4,0,\infty\}$ in $\mathrm{PGL}_2$ . Furthermore, one shows that there exists an absolute constant $M\geq 1$ such that

$|C(K_3,\gamma)|\leq Mp^{1/2}$

for all $\gamma\notin D$ . But $D$ is not upper-triangular, and hence the first argument we sketched is also insufficient here…

I will stop here for now; in the next post, we have two obvious questions to discuss: (1) how does one deal with the complications which arose for $K_2(n)$ and $K_3(n)$ ? and (2) even in the most favorable case, how exactly can one show that the correlation sums are small? As can be expected already from today’s discussion, it is the Riemann Hypothesis over finite fields which is crucial. In fact, to attain the generality of our results, we need to apply it twice; in both cases we require the very deep statements proved by Deligne in the amazing achievement which people call Weil 2, but in one case, this is encapsulated in the Deligne–Katz–Laumon theory of the $\ell$ -adic Fourier transform…

Algebraic twists of modular forms, I

In the next few posts, I want to describe some aspects of the paper “Algebraic twists of modular forms and Hecke orbits” that I just finished with Étienne Fouvry and Philippe Michel. Today, I will describe some cases of the main result (I was thinking of continuing dans la foulée with a high-level sketch of the proof, but it seems better to delay this until the next time…)

I’ll start with one of the simplest cases, which was also one of our original challenges. Let $f$ be a classical holomorphic modular form, say of level $N$ and weight $k\geq 2$ , with Fourier expansion

$f(z)=\sum_{n\geq 1}{n^{(k-1)/2}\rho_f(n)e(nz)},\quad\quad\quad\text{ where } \quad e(z)=e^{2i\pi z}$

for $z$ in the upper half-plane. If in doubt, you may assume that

$f(z)=e(z)\prod_{n\geq 1}{(1-e(nz))^{24}}$

is the Ramanujan $\Delta$ -function, with $N=1$ and $k=12$ .

The (normalized) Fourier coefficients $\rho_f(n)$ are essentially bounded (they are bounded in mean-square norm over $n\leq x$ , by Rankin-Selberg theory) and their signs oscillate quite randomly. In particular, one expects that sums of the type

$\frac{1}{x}\sum_{n\leq x}{\rho_f(n)K(n)}$

should be rather small, unless $K(n)$ is itself somehow related to $\overline{\rho_f(n)}$ .

Our question was: can we prove that this is the case when we take $x=p$ , a prime number, and

$K(n)=e\Bigl(\frac{\bar{n}}{p}\Bigr),\quad\quad\quad K(p)=0,$

(where we denote by $\bar{n}$ the multiplicative inverse modulo $p$ of some integer $n\geq 1$ coprime with $p$ )?

One can indeed ask this as a challenge: considering that we feel that we know something about Fourier coefficients of modular forms, shouldn’t we be able to prove that they do not correlate with $e(\bar{n}/p)$ ? (In addition, there is a geometric motivation, which I will defer to another post.)

It is a very simple and special case of our results that, indeed, we can do something with such sums. Precisely, we prove that

$\sum_{n\leq p}{\rho_f(n)e\Bigl(\frac{\bar{n}}{p}\Bigr)} \ll p^{15/16+\varepsilon}$

for any prime $p$ and for any $\varepsilon>0$ , where the implied constant depends on $f$ and $\varepsilon$ .

The more general statements are all of the following type: for a prime $p$ , we have some “weight” $K(n)$ defined for $n$ modulo $p$ , which we extend to all of $\mathbf{Z}$ by composing with reduction modulo $p$ . Then we want to bound from above the sum

$\sum_{1\leq n\leq p}{\rho_f(n)K(n)},$

for a fixed $f$ . Our results are of the form

$\sum_{1\leq n\leq p}{\rho_f(n)K(n)}\ll \|K\|_{alg}\ p^{15/16+\varepsilon},$

where the implied constant depends only, as before, on $f$ and $\varepsilon>0$ , and where the dependency on $K$ is contained entirely in the quantity $\|K\|_{alg}$ , which I will only define precisely in later instalments, but which roughly speaking measures how difficult it is to express $K$ as a linear combination of special coefficients which I will call here simply “of algebraic origin”.

In fact, we do not express the result in this manner. A technical difference is that we use smoothed sums, proving bounds like

$\sum_{n\geq 1}{\rho_f(n)K(n)V(n/p)}\ll \|K\|_{alg}\ p^{7/8+\varepsilon},$

where $V$ is a smooth function compactly supported on $[0,1]$ and the implied constant now also depends on $V$ , but in very controlled ways. Moreover, we concentrate on the special coefficients of algebraic origin, leaving the general case to the triangle inequality. For instance, our previous weight

$K(n)=e\Bigl(\frac{\bar{n}}{p}\Bigr),\quad\quad\quad K(p)=0,$

is of algebraic origin for every prime $p$ , with $\|K\|_{alg}$ absolutely bounded, and hence the first result is a consequence of the more general case. Another more general type of examples (which remains far from the most general!) is given by

$K(n)=\chi(\phi_1(n))e\Bigl(\frac{\phi_2(n)}{p}\Bigr)$

for integral polynomials $\phi_1$ , $\phi_2$ , and for a Dirichlet character $\chi$ modulo $p$ . All of these are “of algebraic origin”, and when $(\phi_1,\phi_2)$ are fixed, the norm $\|K\|_{alg}$ is uniformly bounded as $p$ varies. Note in particular that $K(n)=\chi(n)$ is possible, and it is certainly among the most natural examples. Indeed, the smoothed form of our bounds leads by classical means to a subconvex estimate for twisted L-functions, which is of the type

$L(f\otimes\chi,1/2)\ll p^{7/8+\varepsilon}.$

This is not new, but it illustrates the range and depth of our results. (Such a bound, with a weaker exponent, was first proved by Duke-Friedlander-Iwaniec, and the exponent $1-1/8$ was first obtained by Blomer and Harcos, refining a method of Bykovski.)

Before finishing for today, here is one example that suggests how one may go well beyond those previously considered. For $p$ prime and $a$ , $b$ coprime with $p$ , let

$S(a,b;p)=\sum_{x\in\mathbf{F}_p^{\times}}{e\Bigl(\frac{ax+b\bar{x}}{p}\Bigr)}$

be the classical Kloosterman sum. Then the weights

$K_1(n)=-\frac{S(n,n;p)}{\sqrt{p}},\quad\quad K_2(n)=\frac{S(n,n;p)^2}{p}-1,$

are “of algebraic origin”. Applying our methods to the second one, we end up with the question discussed in the previous post…

A bijective challenge

Étienne Fouvry, Philippe Michel and myself have just finished a new paper, which is available on my web page and will soon be also on arXiv. This was quite an extensive project, which also opens many new questions. I will discuss the general problem we consider, and the techniques we use, in other posts, but today I want to discuss a by-product that we found particularly nice (and amusing). It can be phrased as a rather elementary-looking challenge: given a prime number $p$ , and an element $t$ of $\mathbf{Z}/p\mathbf{Z}$ which is neither $0$ , $4$ or $-4$ modulo $p$ , let $N_p(t)$ be the number of solutions $(x,y)\in (\mathbf{Z}/p\mathbf{Z}-\{0\})^2$ of the congruence

$x+\frac{1}{x}+y+\frac{1}{y}+t=0.$

The challenge is to prove, bijectively if possible, that

$N_p(t)=N_p\Bigl(\frac{16}{t}\Bigr)$

and that

$N_p(t)=N_p\Bigl(\frac{4t-16}{t+4}\Bigr)=N_p\Bigl(\frac{4t+16}{t-4}\Bigr)$

if $p\equiv 1\pmod{4}$ .

This sounds simple and elegant enough that an elementary proof should exist, but our argument is a bit involved. First, the number $N_p(t)$ is the number of points modulo $p$ of the curve with equation above, whose projective (smooth) model is an elliptic curve, say $E_t$ , over $\mathbf{F}_p$ . Then we checked using Magma that $E_t$ and $E_{16/t}$ are isogenous over $\mathbf{F}_p$ , and this is well-known to imply that the two curves have the same nunmber of points modulo $p$ . The other two cases are similar, except that for

$\gamma(t)=\frac{4t-16}{t+4}\text{ or } \frac{4t+16}{t-4},$

the relevant isogenies are between $E_t$ and $\tilde{E}_{\gamma(t)}$ , where $\tilde{E}_t$ denotes the quadratic twist of $E_t$ by $-1$ . Hence the number of points are the same when $-1$ is a square modulo $p$ .

In the first case, the isogeny is of degree $4$ , and the others are of degree $8$ , so the formulas which define them are rather unwieldy, at least in the equivalent Weierstrass model.

The best explanation of this has probably to do with the relation between the family of elliptic curves and the modular curve $Y_0(8)$ (a relation whose existence follows from Beauville’s classification of stable families of elliptic curves over $\mathbf{P}^1$ with four singular fibers, as C. Hall pointed out), but we didn’t succeed in getting a proof of all our statements using that link. In fact, we almost expected to find the identities above already spelled out in some corner or other of the literature on modular curves and universal families of elliptic curvers thereon, but we did not find anything.

In the next post, I’ll come back to this to explain the link with our paper, which ostensibly is about estimates for sums of Fourier coefficients of modular forms multiplied with functions “of algebraic origin”. Kloosterman sums will enter the picture to make the connection (in more ways than one!), and we’ll see a rather elegant formula of Beltrami…

P.S. Here is a link to a transcript of a Magma session proving the existence of the isogenies which imply our formulas, and ending with the formula of the $4$ -isogeny, written in terms of the original curve.