Mathematics – Page 21 – E. Kowalski's blog

Local limit theorems slides

I gave today a lecture in the conference in honor of F. Delbaen at ETH, and since the rooms were not suitable for blackboard talks, I prepared a beamer talk. I’ve put up the slides on the web for any interested reader; the topic was a survey of the general ideas surrounding my papers with A. Nikeghbali, J. Jacod, A. Barbour, and F. Delbaen on mod-gaussian convergence, mod-Poisson convergence, and related limiting behavior of sequences of random variables. I have posted about this a few times before, but not about all of the results. All the corresponding papers and preprints can be found on my home page.

Correlation sums in the wild

In my last post concerning my joint work with É. Fouvry and Ph. Michel, I reported a few weeks ago how happy we were to have found in the literature a specific case of the general correlation sums that we introduced in our paper to deal with “algebraic twists” of modular forms. The example, we are happy to report, turns out not to be isolated: we have found three more in the last few days. I only list them in the briefest way below, since some of them are rather complicated looking, but precise statements and references are found in a short note we just finished typing. It is rather nice to see how some order emerges from these sums (the last one has no less than 8 parameters, in addition to the three variables of summation) once they are considered from the point of view of general correlation sums.

In a paper from 1990 concerning small eigenvalues of the hyperbolic Laplace operator in special situations, H. Iwaniec considers correlation sums related to the weight $K(x)=e(2\bar{x}/p)S(\bar{x},\bar{x};p)$ , where $\bar{x}$ is the inverse of $x$ modulo $p$ , and $S(m,n;p)$ is the usual Kloosterman sum; the matrices $\gamma$ in the correlation sums are here all upper-triangular, the difficult case being when $\gamma$ is not diagonal. It turns out that the general machinery we develop proves the desired estimates for these sums.
In a paper of 1995, N. Pitt considers correlation sums related to the weight $K(x)=e(\bar{x}/p)$ , which was also the one involved in the sums of Friedlander and Iwaniec that we had earlier identified as examples of correlation sums. However, whereas the matrices in that first case were lower-triangular, there is no particular restriction on the sums of Pitt (which are somewhat involved, with 4 parameters), except that they are not upper-triangular.
Finally, in a recent preprint, R. Munshi also considers correlations sums related to $K(x)=e(\bar{x}/p)$ , also without any particular restriction on the matrices involved except that they are not upper-triangular. These sums differ from those of Pitt by the number and configuration of parameters (there are 8 here…)

We have not yet fully updated the text of the paper to mention these examples, but this will be done soon…

Lecture notes list

Just a note to mention that I’ve just created a web page with links to the notes I’ve written for various courses at ETH over the years (though most of them are not as complete as I would like).

An exponential sum of Conrey and Iwaniec

In their remarkable paper on the third moment of special values of twisted $L$ -functions of modular forms, Conrey and Iwaniec require (among many ingredients!) a best-possible estimate (square-root cancellation) for the exponential sums
$S(\chi,\eta)=\sum_{u,v}{\chi(uv(u+1)(v+1))\eta(uv-1)}$
where $\chi$ and $\eta$ are non-trivial multiplicative characters modulo a prime $p$ . Using a combination of the formalism of such exponential sums (including the consequences of the rationality of the corresponding $L$ -functions over finite fields and of the Riemann Hypothesis of Deligne) together with some elementary averaging arguments inspired by ideas of Bombieri, and special treatment of the case $\chi=\eta$ , they succeed in proving that
$S(\chi,\eta)\ll p,$
which is the expected estimate.

I looked at this sum recently because I wondered if (like the other sum of Friedlander and Iwaniec mentioned at the end of my previous post) it would turn out to be a special case of the correlation sums that feature in my recent paper with Fouvry and Michel. As far as I can see, it isn’t of this type, but this led me to attempt to find a proof of the estimate of Conrey and Iwaniec based on the philosophy of reducing a character sum in more than one variable to a one-parameter sum which involves more complicated summands than character values.

This does indeed work. I’ve written a short note explaining this (it is not intended for publication, so rather terse at some points). The outline is rather straightforward: we write
$S(\chi,\eta)=\sum_{u}R(u)T(u)$
where
$R(u)=\chi(u(u+1))$
is a character and
$T(u)=\sum_{v}{\chi(v(v+1))\eta(uv-1)},$
which is a more “complicated” summand, but still a one-variable character sum.

It turns out that $R(u)$ and $-T(u)$ are both among these nice algebraic trace functions that occur in the paper with Fouvry and Michel. In particular they have the crucial irreducibility property that make such functions quasi-orthogonal, which means here that we get
$\sum_u R(u)T(u)\ll q$
unless $\overline{R(u)}$ and $T(u)$ are proportional for all $u$ . (The size $q$ on the left-hand side comes from the fact that $T(u)$ is of weight $1$ , i.e., a sum of a bounded numbers of algebraic numbers of modulus $\sqrt{q}$ ; the one-variable sum over $u$ gives square-root cancellation — by the Riemann Hypothesis — for the bounded summands $q^{-1/2}R(u)T(u)$ .)

One should feel that it is unlikely that $\overline{R(u)}$ and $T(u)$ are proportional, and indeed it is easy to show during the construction that they are not (the algebraic reason is that one is the trace of a representation of degree $1$ , whereas the other is a trace of a representation of degree $2$ …). To finish the proof, one must still be careful to control the implied constant in the estimate above, since it is a priori dependent on the characteristic of the finite field involved. A true dependency like this would be catastrophic, but one can show (in different ways) that this constant is in fact bounded uniformly (the basic reason is that the invariants measuring the complexity of the functions $R(u)$ and $T(u)$ are bounded independently of $p$ ).

This argument is quite clean in outline (and requires no special consideration of special cases like $\chi=\eta$ ) but it uses rather deep tools. In addition to three applications of the Riemann Hypothesis (one to understand the summand $T(u)$ , another to perform the sum over $u$ , and the last as explained in a second), there is a fair amount of deep formalism involved in checking that $T(u)$ has the desired properties for a nice algebraic function. There is one last nice ingredient: in order to ensure quasi-orthogonality, one needs to know that $T(u)$ is irreducible in some precise sense. This requirement turns out to translate (using a last time the Riemann Hypothesis) to a nice diophantine property of these sums, namely that
$\lim_{|k|\rightarrow +\infty}\frac{1}{|k|^2}\sum_{u}{|T_k(u)|^2}=1,$
where $k$ runs over finite extensions of $\mathbf{F}_q$ and, denoting by $N$ the form from $k$ to $\mathbf{F}_q$ , the sum
$T_k(u)=\sum_{v\in k}{\chi(N(v(v+1)))\eta(N(uv-1))},$
is the natural companion of $T$ to such extensions. The proof of this limit formula is a nice exercise in manipulating finite sums in the tradition of analytic number theory…

Algebraic twists of modular forms, III

After a small break while traveling, I continue the discussion of my paper with Fouvry and Michel, continuing from here. But first of all, I’m happy to mention two proofs of the counting identities mentioned in my introductory post: one that D. Zywina has sent me, a nice proof based on explicit computations of modular forms and functions, and another that is in fact just an application of a more general result of Deligne and Flicker on local systems on the projective line minus four points (see Section 7 of this preprint).

Now, if memory serves, at the end of the last post, we saw that analytic techniques reduce the study of our sums

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

to the study of correlation sums

$C(K;\gamma)=\sum_{z}\overline{\hat{K}(\gamma\cdot z)}\hat{K}(z),$

where $\hat{K}$ is the normalized Fourier transform modulo $p$ .

For these, we need to show that “most” are small in order to conclude. Fixing a parameter $M\geq 1$ , let us therefore define $G_{K,M}$ to be the set of matrices $\gamma$ in $\mathrm{PGL}_2(\mathbf{F}_p)$ such that

$|C(K;\gamma)|>Mp^{1/2}.$

The discussion in the previous post shows that if, for some $M\geq 1$ , all matrices in $G_{K,M}$ are upper-triangular, we get

$S(f;K)\ll p^{1-1/8+\epsilon}$

for any $\epsilon>0$ . But we can deal with some more exceptions. More precisely, we show that this estimate is still valid, with an implied constant depending only on $M$ , if the matrices in $G_{K,M}$ are all either

parabolic (they have a single fixed point in $\mathbf{P}^1$ ), or
upper-triangular, or
if they either fix or permute two distinct points taken in a finite list, say containing at most $M$ pairs $\{x,y\}$ .

Compared with the previous discussion, the subtlety here is that it can indeed happen that such matrices appear in the transformations we apply to $S(f;K)$ , and proving that their contributions remain under control involves some rather fun analysis.

This summarizes, in a rather hurried fashion, the first part of our paper. Logically, we obtain statements which are self-contained, but which are only applicable directly in a few cases (the case $K(n)=\chi(n)$ is an excellent example where this can be done). To go further, we need to use algebro-geometric tools.

And here comes a dilemma well-known to anyone who has had to present research involving two relatively distant areas of mathematics, so that specialists of one may not know the other. I can give a concise definition of the class of weights $K(n)$ that we consider, and it will be immediately familiar and natural to algebraic geometers — but not to most analytic number theorists.

So, instead, I will just say (very fast) that these weights, which we call “irreducible trace weights”, are trace functions of geometrically irreducible $\ell$ -adic middle-extension Fourier sheaves pointwise pure of weight $0$ on $\mathbf{A}^1/\mathbf{F}_p$ . Then I will defer to a later post a more leisurely run-through this definition, together with more examples of weights and their formalisms, and with some further analytic properties which are of independent interest.

The reason this class of weight “works” can however be quickly summarized in a rather miraculous property, which is essentially a consequence of the Riemann Hypothesis: assuming $K$ has small complexity (in a precise sense, saying basically that it comes from a sheaf with small rank and small ramification), the correlation sum $C(K;\gamma)$ is either $\ll p^{1/2}$ , where the implied constant depends only on the numerical invariant measuring the complexity of the weight, or we have an equality

$\hat{K}(z)=\varepsilon(\gamma) \hat{K}(\gamma\cdot z)$

for all $z$ and some fixed complex number $\varepsilon(\gamma)$ of modulus $1$ . From this second part, we see that, for a suitable $M$ , the set $G_{K,M}$ discussed above is contained in the group $\mathbf{G}_K$ of all $\gamma$ (viewed in $\mathrm{PGL}_2(\mathbf{F}_p)$ ) for which there exists $\varepsilon(\gamma)$ (of modulus $1$ ) such that the identity above holds. This provides us with enough structure on the set of “bad” matrices (with large correlation sums), from which the bound

$S(f;K)\ll p^{1-1/8+\epsilon}$

can fairly simply be deduced for irreducible trace weights using the automorphic part of our paper.

Indeed, we distinguish two cases, depending on the structure of the subgroup $\mathbf{G}_K\subset \mathrm{PGL}_2(\mathbf{F}_p)$ :

If $\mathbf{G}_K$ has order coprime to $p$ , we use the classification of such subgroups, and see that $\mathbf{G}_K$ is either contained in the normalizer of some maximal torus, i.e., in the stabilizer of a two-point set $\{x,y\}$ (and hence these weights fall under the third case (3) of allowed “bad” matrices) or otherwise $\mathbf{G}_K$ is of order at most $60$ , and consists only of semisimple elements (which allows us again to apply (3), with possibly more than one pair $\{x,y\}$ involved, but less than $60$ );
If $p\mid \mathbf{G}_K$ , we find some element $\gamma_0$ in $\mathbf{G}_{K}$ which is of order $p$ , hence unipotent. Thus $\gamma_0$ acts transitively on $\mathbf{F}_p$ (minus at most a single point); the formula defining the subgroup $\mathbf{G}_K$ above implies easily that $\hat{K}(z)$ is then basically of a very restricted type, namely
$\hat{K}(z)=\varepsilon_0 e(a\gamma\cdot z/p)$
for some fixed matrix $\gamma$ and fixed complex number $\varepsilon_0$ and integer $a$ . But this weight comes from a specific (Fourier transform of a) Artin-Schreier sheaf (it might not be the one defining $K$ originally, but a fortiori, we can assume it is!). For this sheaf, a rather simple analysis shows that $\mathbf{G}_K$ is a unipotent group isomorphic to $\mathbf{F}_p$ (unless $a=0$ or $\gamma=1$ , which are exceptional cases). So the bad matrices are either trivial or parabolic, and we can appeal to case (1) to handle these weights…

I will conclude for today with another fact we noticed only very recently: in the special case $K(n)=e(\bar{n}/p),$ it turns out that the correlation sums had already appeared in a paper of Friedlander and Iwaniec on incomplete Kloosterman sums, in the special case of $C(K;\gamma)$ when $\gamma$ is lower-triangular (and non-diagonal). In the Appendix to this paper, Birch and Bombieri give two proofs of the estimate $C(K;\gamma)\ll p^{1/2}$ (for lower-triangular matrices): one is geometric (based on counting points on surfaces over finite fields), but the second one has amusing links to our arguments, with a camouflaged sighting of the group structure of the group $B_-$ of lower-triangular matrices and of the fact that $G_{K,M}\cap B_-$ is contained in a subgroup of $B_-$ … (Interestingly, there is no trace of modular forms in this paper of Friedlander and Iwaniec, so the coincidence is rather unexpected.)