Notes and unpublished results by E. Kowalski

Here are, on the one hand, short notes which were written mostly to record various facts or as answer to questions raised by other people, and on the other hand some rather longer unpublished papers and surveys, including the original version of my Ph.D. thesis.
They are in more or less reverse chronological order, with an approximate date indicating when the current state was achieved. Note that no particular claim of correctness is made, although each note has received a decent amount of attention.
I mention every prior work that I am aware of. Some of the results turned out to be known before I figured out the answer independently, and others may well be already in the literature (some are consciously reports on other people's work, and identified as such); of course, I will be happy to cite any related work that is pointed out to me. In particular, finding a statement with no attribution in one of the notes below does not imply that I claim priority.
Fourier expansion of Schoenberg's space-filling curve. September 2019
We record the computation of the Fourier expansion of the space-filling curve defined by Schoenberg, and check that its Fourier coefficients grow too fast to provide an example of a space-filling curve in the support of the Kloosterman path (see note 41 below).
Complements to Fouvry-Katz-Laumon stratification. 2015; january 2019
This is a complement to the stratification results for exponential sums of Fouvry-Katz and Katz-Laumon, where the underlying variety is allowed to vary in a family, and we look for uniform descriptions of the strats.
On the support of the Kloosterman paths, with W. Sawin. 2017; january 2019
We compute the support of the Kloosterman paths (as defined in the paper Kloosterman paths and the shape of exponential sums) and derive some consequences concerning the symmetries of the partial sums of Kloosterman sums; we also consider whether various classical functions of real analysis belong to the support. I have written some blog posts about these notes. Unless and until the most interesting open question (on the existence of a space-filling curve in the support) is settled, these notes will not be submitted for publication.
The geometric Bunyakowsky problem 2016; january 2019
I discuss a geometric problem concerning the Galois group of coverings of function spaces on curves obtained by intersecting the graph of the functions with a fixed curve. In the case of polynomials of fixed degree over a finite field, the computation of this group leads to an asymptotic formula, as the size of the field increases, for the number of irreducible values F(f) for a polynomial F in two variables, which is the "large finite field" case of Bunyakowsky's Conjecture. Using Lefschetz pencils on curves and elementary group theory, we prove that the Galois group is as large as possible in many cases (in the classical case, such computations have been done by Hall, Pollack, Bary-Soroker and especially Entin).
Warning! The algebraic geometry in positive characteristic has some mistakes, pointed out by W. Sawin. So the strategy is somewhat sound, but the actual results are not correct. However, the notes will probably remain in the current state (the paper was never submitted for publication).
39 Unmotivated ergodic averages January 2019, September 2023
These notes present some statements related to correlations between trace functions and orbit sequences. Although it turns out that the problem is somewhat disappointing (as are the proofs), the latest version might be of some minor interest (in particular, it makes use of an apparently new statement of diophantine approximation with "chosen" numerators, which will be posted separately), and the text is posted for this reason.
38 A Fourier theoretic criterion for convergence in law June 2018
This is a brief note containing a criterion for convergence in law in the Banach space of continuous functions on an interval, where convergence of Fourier coefficients replaces the usual requirement of convergence of finite distributions. This may be easier to check when the Fourier coefficients satisfy some independence properties.
37 Sahakian's Theorem and the Mihalik-Wieczorek problem May 2018
This note checks formally a statement mentioned in a blog post: if there exists a continuous function on [0,1] such that the image of any interval is a convex set, and such that the image is not contained in a line, then there also exists a space-filling curve with Fourier coefficients decaying like 1/|h|. It is maybe more interesting to view this as a detailed account of Sahakian's Theorem on reparameterization of Fourier series...
36 Symmetric powers and the Chinese restaurant process November 2016
We explain how the computation of the dimension of symmetric powers of vector spaces leads to a proof of the "Chinese restaurant" decomposition of the number of cycles of random permutations in terms of a sum of independent Bernoulli random variables.
35 Non-multiplicativity of Fourier coefficients of modular forms for non-arithmetic groups: a theorem of A. Venkatesh June 2016
This quick note records the proof by A. Venkatesh of the fact that Fourier coefficients of holomorphic cusp forms for non-arithmetic groups are non-multiplicative in some precise (but rather qualitative) sense. This is motivated by a result of Duke and Iwaniec that shows the same property for half-integral weight forms for congruence subgroups (in a much more quantitative manner).
34 Sums of A. Irving, with É. Fouvry and Ph. Michel. March 2015
We explain how to apply the methods described in our paper A study in sums of products to estimate a family of exponential sums introduced by A. Irving in his paper Estimates for character sums and Dirichlet L-functions to smooth moduli (this appears also as an Appendix to that paper).
33 The Friedlander-Iwaniec sum, with É. Fouvry and Ph. Michel. May 2013
In view of the interest in the exponential sums of Friedlander and Iwaniec due to their crucial appearance in the work of Yitang Zhang on bounded gaps between primes, we spell out briefly the conceptual argument for their estimation that arises naturally from seeing them as examples of correlations sums, and from the application of the results in our paper Algebraic twists of modular forms and Hecke orbits.
32 Correlation sums in the wild, with É. Fouvry and Ph. Michel. September 2012
This note contains the up-to-date list of all instances of "correlations sums" (as defined in our paper Algebraic twists of modular forms and Hecke orbits) that we are aware of in the literature. As of 21.9.2012, this list contains sums of Friedlander and Iwaniec, Iwaniec, Pitt and Munshi (in chronological order).
31 On some exponential sums of Conrey and Iwaniec August 2012
This note outlines a proof of an estimate of Conrey and Iwaniec for a certain character sum in two variables which is used in their paper on the third moment of special values of twisted automorphic L-functions. The argument is based on the principle of reducing to one-variable sums with summands involving trace functions of suitable sheaves, and then applying Deligne's general formalism of the Riemann Hypothesis for such sheaves.
30 Explicit multiplicative combinatorics October 2011
This is a very condensed account, written for reference purpose, of following the standard results (due in particular to Balog-Gowers-Szemerédi and Tao) on products sets in finite groups to obtain explicit constants. In particular, this is used in my lecture notes on expander graphs to obtain explicit estimates for the Bourgain-Gamburd argument for expansion, where one needs to relate sets with large "multiplicative energy" to "approximate groups". The arguments in this note will be incorporated, and expanded in a more accessible manner, in the lecture notes.
29 Note on an inequality of Burnside May 2011
This note discusses, from the point of view of minima of integral quadratic forms, an inequality of Burnside about representations of finite cyclic groups, which comes up in proving that an irreducible complex representation of degree at least 2 of a finite group has at least one zero. This involves minima of tensor products of quadratic forms, a rather fascinating topic where the names of Siegel, Conway, Thompson, Steinberg, Milnor and Husemoller appear.
28 Sur les zéros des fonctions L automorphes de grand niveau, with Ph. Michel 1997; December 2010
This is a very old preprint (Prépublication Mathématique de l'université d'Orsay, number 54, 1997) written with Ph. Michel. Most of the results contained in it were published, usually in stronger forms (especially in the papers 2 and 3 on my list of papers, and in my PhD thesis 3 on this page), and many have since been improved and generalized. However, the proof and statement of one particular density theorem was never published (Theorem 1.1 in the preprint).
27 On the complexity of Dunfield-Thurston random 3-manifolds September 2009, September 2010
This short note explains, on the one hand, how to improve the lower-bound for the "typical" size of the torsion subgroups of the first homology of Dunfield-Thurston 3-manifolds (see note 14 below), and also howto give a partial intrinsically geometric interpretation of the length k of the random walk which is involved, by relating it to the "complexity" of the manifold, as defined by Matveev.
26 Reading Burnside September 2010
This is an attempt to interpret, in modern language, Burnside's proof of his theorem that the group algebra of an irreducible linear group over an algebraically closed field is the full endomorphism algebra of the underlying vector space. This proof turns out to be very well motivated and quite short; it is also a good example of applications of elementary ideas of representation theory.
25 Most hyperelliptic curves have big monodromy March 2008, August 2010
This is a self-contained version of an Appendix I wrote for a paper of C. Hall, "Transvections and l-torsion of abelian varieties". He describes a sufficient condition to ensure that, for all but finitely many primes, the Galois group of the l-torsion points of an abelian variety over a number field are "as big as possible", and this note shows that this condition is satisfied for almost all jacobians of hyperelliptic curves (in some sense). The basic tool is an application of sieve techniques to ensure that his conditions are satisfied; one of them also requires a nice result of Zarhin determining the endomorphism ring of certain jacobians.
24 Mod-Cauchy convergence February 2010
This short note records three instances of "mod-Cauchy convergence", the analogue for the family of Cauchy distributions of the notions of mod-Gaussian and mod-Poisson convergence (discussed in papers numbered 35 and 38 on my papers page). These are re-interpretations of results of Vardi on Dedekind sums, of Sarnak on linking numbers of modular knots, and of Spitzer on the winding number of complex Brownian motion. See this blog post, or that one for some more discussion.
23 A combinatorial "intermediate value" lemma April 2001, November 2008
The lemma in question states that, given a positive integer m, one can find another positive integer N such that, among any choice of N integers between 1 and m, there is a subset whose sum is equal to N. In fact (except possibly for m=5 or 6), N can be taken to be the least common multiple of the integers from 1 to m. I had asked this question (around March, 2001) to L. Habsieger, and this note is my recent English adaptation of an earlier write-up of his proof of this lemma, which he found around the same time.
22 An alternate argument for the arithmetic large sieve inequality September 2008
This short note describes a very natural and well-motivated derivation of the "arithmetic" large sieve inequality from the dual of the analytic inequality, which avoids the usual trick of submultiplicativity of Gallagher. This is also described in a blog post and is incorporated in the paper Amplification arguments for large sieve inequalities.
21 Modular signs, or yet another recognition problem for modular forms April 2008
As described in a blog post, this note explains how to prove that two primitive modular forms for which the signs of the Fourier coefficients coincide must be equal, by working around the lack of general proof of the Sato-Tate Conjecture. A stronger version of this result (based on an approximation to the "pair-Sato-Tate conjecture") is included in the latest version of the paper Modular signs (joint with Y-K. Lau, K. Soundararajan and J. Wu).
20 The ubiquity of surjective reduction in random groups October 2007
This very short note is another application of the large sieve for discrete groups (which this time will not be included in The large sieve and its applications); using results of Kantor of Lubotzky (which depend on the classification of finite simple groups...), it follows straightforwardly that a subgroup of SL(n,Z) generated by two "random" elements has probability tending to 1 to surject modulo p for some pretty small prime p, as the complexity of the elements increases.
19 Pointwise bounds for orthonormal basis elements in Hilbert spaces September 2007
This short note considers, in some cases, the question of finding orthonormal basis of a Hilbert space of functions with minimal L norm. This again may be well-known but I haven't found a reference. The optimal bound is found for finite-dimensional spaces; and it is explained that most infinite dimensional examples behave very differently: it is then often possible to find an orthonormal basis where all functions have constant modulus 1 (assuming the inner-product is normalized so that such functions have norm 1), whereas such a behavior characterizes the "uniform density" case for a finite set.
18 A funny identity July/October 2007
This contains simply an easy proof of a funny combinatorial/binomial identity, which may be used to recover directly the Keating-Snaith formula for integral moments of characteristic polynomials of unitary matrices from the probabilistic interpretation of the Haar measure on SU(N) due to Bourgade, Hughes, Nikeghbali and Yor (see arXiv:0706.0333, which uses hypergeometric identities to derive directly the more general complex moment formula).
17 Splitting fields of characteristic polynomials in algebraic groups August 2007
This is again (more or less) a self-contained extract of The large sieve and its applications. The goal is to give an intrinsic general proof of the fact that the Galois group of the splitting field of the characteristic polynomial of a matrix in a reductive linear algebraic group can be identified with a subgroup of the Weyl group. This is a trivial fact for GL(n), and is easy to check for Sp(2g) using the "functional equation" of the characteristic polynomial, but it seems interesting to have a general argument.This is also related with recent results of Corvaja (see arXiv:math/0610661v2).
16 Excluding certain bad behavior of Fourier coefficients of modular forms May 2007
This very short note shows that a certain very biased type of behavior of Fourier coefficients can not occur too often for holomorphic primitive cusp forms of even weight. The case of Maass forms, which could have applications, seems much harder since we use some algebraic properties of the Hecke eigenvalues. The type of bias which is considered occurs in Holowinsky's approach to Quantum Unique Ergodicity.
15 The principle of the large sieve October 2006
This is the original preprint version of the book The large sieve and its applications, and can serve as a survey for the main results and techniques (although there are some inaccuracies which are fixed in the book draft but haven't been incorporated back here...). See also arXiv:math.NT/0610021.
14 The large sieve, property (T) and the homology of Dunfield-Thurston random 3-manifolds December 2006
This note contains some further applications of the general form of the large sieve developped in the preprint The principle of the large sieve, and in the forthcoming book The large sieve and its applications. Those applications concern the notion of random 3-manifolds studied in a recent work by N. Dunfield and W. Thurston; precisely, some of their results are refined by giving strong quantitative upper bounds for the probability that such a random manifold has positive first Betti number, and lower bounds with high probability for the size of the integral homology.
13 The elliptic sieve August 2006
This note gives an amusing application of the "dual" form of the large-sieve inequality (going back to Renyi) to prove that the denominators of most rational points on an elliptic curve have many prime factors. This is also related to similar questions concerning so-called elliptic divisibility sequences.  As is the case for the previous note, this result is incorporated in the preprint The principle of the large sieve, and in the forthcoming book The large sieve and its applications.
12 Bounds for degrees and sums of degrees of irreducible characters of some classical groups over finite fields June 2006, February 2011
This note explains how to prove fairly sharp explicit upper bounds for the maximal degree, and the sum of the degrees, of the irreducible representations of some finite groups of Lie type; the techniques for this are based on Deligne-Lusztig characters and were explained to me by Jean Michel.  The results were incorporated in the preprint The principle of the large sieve and in the book The large sieve and its applications.
In late 2010, from a preprint of Larsen, Malle and Tiep, I learnt that the bound for the maximal degree was in fact known before. Indeed, it had been proved by G. Seitz in 1989, in a more general form. I have added references to these papers; the proof here is a bit different -- and maybe a little bit more elementary -- from the one of Seitz, though both are based on Deligne-Lusztig theory.
11 Big symplectic monodromy: a theorem of C. Hall May 2006
This note is simply a rephrasing of a special case of results of C. Hall that can be used to show that certain geometric monodromy groups modulo ell are as large as possible; specifically, it explains the symplectic case and how it yields an alternate proof of a theorem of J-K. Yu used in my paper The large sieve, monodromy and zeta functions of curves.
10 A two-dimensional larger sieve March 2006
This tries to get some higher-dimensional version of the larger sieve of Gallagher; this brings out some interesting points, but it turns out that in a straightforward adaptation at least, this only works for the same range of number of permitted residue classes (not density of residue classes).
9 Sommes de deux carrés successifs qui sont des carrés November 2005
This explains that there are infinitely many integers such that n2+(n+1)2 is a square (I have no memory why I was interested in this question...). The values of n form sequence A001652 of the Online Encyclopedia of Integer Sequences.
8 Counting primes irrationnally October 2005
This explains how the fact that ζ(2) is irrationnal "implies" that π(x) is at least of the order of log log x... (or at least that there are infinitely many primes; this is something I heard from H. Iwaniec). More refined results are in a preprint of Miller, Schiffman and Wieland (see arXiv:0709.2184), and this was also observed by J. Sondow (see arXiv:0710.1862).
7 Les entiers de la forme a2+mb2May 2004
This explains how to find the asymptotic formula for the number of integers of this type (without multiplicity). This was written to answer a question of Fouvry, before we became aware that this had been solved by Bernays in the early 20th century. V. Blomer has recently obtained much more precise results.
6 The global root number for J0(q) March 2003
This relates the global root number for the jacobian of the modular curves with q prime to class numbers of imaginary quadratic fields modulo 4. In particular, it is not known whether this global root number is evenly distributed among +1 and -1.
5 Bezout curves in the plane December 2001, February 2019
This unpublished paper studies the following question: if a set in the projective plane over an algebraically closed field has the "same" intersection properties with algebraic curves as an algebraic curve of some degree d, does it follow that the set is itself an algebraic curve? The paper proves at least that any such "Bezout curve" is not Zariski-dense, and in some cases that it is the complement of finitely many points in the Zariski-closure.  There is an amusing link with a "geometric" converse theorem for algebraic curves over finite fields.
Update (Feb. 2019) Will Sawin has sent me a sketch of proof of the question that was left in this note: he shows that a Bezout curve is indeed, up to finitely many points, an algebraic curve. See the Addendum to the paper reproducing his email.
4 Deligne's proof of the Weil conjectures for varieties over finite fields1997/1998
These notes (33 pages) present what I understood of Deligne's (first) proof of the Weil conjectures (for smooth, projective varieties) around 1998/1999. The goal was to explain this as simply as possible, and the first part, in particular, can be interesting for a very first look at étale cohomology (with a more or less complete computation, from scratch, of the cohomology of an elliptic curve, using the basic theory of isogenies).
3 The rank of the jacobian of modular curves: analytic methods April 1998
This is my Ph.D. thesis, defended at Rutgers University in 1998. Most of its contents are found in the papers numbered 1 (with W. Duke and D. Ramakrishnan), 2 and 3 (with P. Michel) on the main page.
2 Théorie d'Atkin-Lehner pour les formes paraboliques de Maass, représentations automorphes de GL(2) 1992
This is the report for my D.E.A (i.e, Master's Thesis), Grenoble 1992.  It contains also the work I did during my Rutgers internship in summer 1991 with H. Iwaniec. Mostly posted for archival reasons...
1 La fonction de Beurling et la grand crible. Applications. 1990
This is the report of my first year internship at ENS Lyon, 1990, written under the direction of E. Fouvry and H. Daboussi.  As the previous item, this is mostly of archival interest for myself...

Last update 15.9.2023 by E. Kowalski