Papers, books, translations, surveys
sums over small subgroups.
This is an account of the proof by Bourgain, Glibichuk and Konyagin of non-trivial bounds for exponential sums over very small multiplicative subgroups of prime finite fields. The argument is written in more explicitly probabilistic style, which may be more intuitive for some readers. The paper also includes a proof of the relevant version of the Balog-Szemerédi-Gowers Theorem.
approximation with chosen numerators.
This is a very short note, which proves the existence of a sequence of integers indexed by primes such that almost all numbers between 0 and 1 are infinitely often at distance at most
(Science China Mathematics, 2023)
coefficients in large arithmetic progressions,
This paper, in the special volume dedicated to J. Chen, concerns the distribution of coefficients of certain Rankin-Selberg L-functions in arithmetic progressions to large moduli. It is an elaboration of the methods and results of the paper 77.
statistics from spectral measures on tensor envelopes,
This short paper introduces a new notion of "spectral measure" associated to objects of a tensor category, which is inspired by the corresponding definition for normal operators acting on Hilbert spaces. We then describe a quite surprising application: we present a new proof using spectral measures of the classical solution of the problème des rencontres, which goes back to the early 18th Century emergence of probability theory in the analysis of card games, and marks the first occurence of the Poisson distribution. The relevant tensor categories are the Deligne-Knop categories of "representations" of the symmetric group St, when t is an indeterminate, and one interpretation of our result is that the Poisson distribution with parameter 1 is a "Sato-Tate measure" for this fantasmatic group. We conclude with some arithmetic speculations related to a hypothetical Galois theory of pseudo-polynomials.
families and averages of L-functions, I,
We initiate the study of families of L-functions related to algebraic subgroups of Dirichlet characters (with prime modulus). As a first concrete instance of these types of problems, we determine the asymptotic behavior, as q prime tends to infinity, of the value L(1/2,χa)L(1/2,χb), for a and b integers.
(Acta Arithmetica, 2023)
sums of trace functions,
This is an exploration of various generalizations of previous results of Duke, Garcia, Hyde, Lutz, and others, which concerned finite sums of additive characters evaluated at roots of unity of a fixed order in finite fields. We show that the equdistribution properties that were observed in these works are valid in many other contexts, in particular when replacing the roots of unity by the zeros of any fixed integral monic polynomial, and including replacing the additive characters with multiplicative characters or with trace functions of certain kinds. The (integral) additive and multiplicative relations satisfied by the roots of a polynomial play a crucial role in determining the limiting distributions.
sets in algebraic geometry,
We show that in most cases, the image of an algebraic curve in a generalized jacobian is a Sidon set, or a symmetric Sidon set. We comment on the structural properties of these sets compared to known dense Sidon sets, in the case of finite fields. In particular, we show that in the case of genus zero, there are five examples which correspond to the five dense Sidon sets classified recently by Eberhard and Manners using projective planes, and we show that genus one curves and genus curves give rise to new families of fairly dense Sidon sets. This article supersedes the paper 83 below.
(Bourbaki seminar, 2022)
additive problems for polynomials over finite fields, following
W. Sawin and M. Shusterman.
This is the written text of my April 2022 talk in the Bourbaki seminar concerning the papers of Sawin and Sawin-Shusterman which establish the twin prime and quadratic Bateman-Horn conjectures, as well as extremely strong level of distribution results, in the context of fixed finite fields, and polynomials of increasing degree.
sums, twisted multiplicativity and moments, with
We study the sums of the modulus of (normalized) exponential sums with polynomial phases, strenghtening and generalizing earlier work of Fouvry and Michel. We also consider the problem of showing non-correlation between sums associated to different polynomials.
The methods combine analytic number theory (related to the previous paper 79) and the algebraic interpretation of exponential sums over finite fields (especially the work of Katz).
Fourier transforms over finite fields: generic
vanishing, convolution, and equidistribution, with
This monograph establishes the foundations for the generalization to all connected commutative algebraic groups of Katz's approach to equidistribution of arithmetic Fourier transforms, based on tannakian categories constructed using the algebraic convolution on the group. We moreover discuss a variety of examples, including improving results of Hall, Keating and Roddity-Gershon on the variance of von Mangoldt functions of higher-degree L>/i>-functions in arithmetic progressions (over function fields).
curves in their jacobians are Sidon sets, with
We observe that an algebraic curves of genus at least two, embedded in its jacobian, is a Sidon set (up to the hyperelliptic involution in the case of hyperelliptic curves); this was known to N. Katz, except for the terminology, and implies that any potential classification of "large" Sidon sets would have to take into account such examples. This paper has been superseded by 87 above (with F. Fresán). above,
(Journal of the AMS, 2022)
Quantitative sheaf theory, by
; mis en forme by
, and myself
This paper develops W. Sawin's complexity theory for (complexes of) ℓ-adic sheaves on quasi-projective varieties, and gives first applications. This theory is a crucial ingredient in the monograph 84 of A. Forey, J. Fresán and myself on equidistribution of exponential sums parameterized by characters of arbitrary connected commutative algebraic groups over finite fields.
(Cambridge Univ. Press, 2021)
An introduction to probabilistic
This is the published book version of my lecture notes on probabilistic number theory. The accompanying web page contains (or will contain...) useful links and a list of corrections.
(Gaceta Real Soc. Mat Esp., in Spanish)
A short survey of probabilistic number theory, written in the style of my forthcoming introductory book above (itself based on my lecture notes). The spanish translation (by J. Fresán) appears in the section "El Diablo de los Números" of the Gaceta de la RSME.
(Advances in Math. 2021, Open Access)
from the Chinese Remainder Theorem, with
We prove that if one is given some subsets of residues classes modulo primes, and extend these to subsets modulo all squarefree moduli using the Chinese Remainder Theorem, then the fractional parts of these sets will become equidistributed modulo 1 if either (1) the sets modulo primes have at least two elements for a positive density of primes, and the moduli have sufficiently many prime factors; or (2) the sets modulo primes has growing size, and the moduli have at least two prime factors.
This puts in a much wider context (and strenghtens) a previous result of Hooley concerning roots of polynomial congruences. We also prove higher-dimensional analogues and consider various applications, e.g., to roots of pseudopolynomials (in the sense of Hall), and to an "all moduli" analogue of a question of Hrushovski.
(Contemporary Math., 2019)
on applied ℓ-adic cohomology, with
, and .
This is a detailed survey of many of the results about, or proved with, trace functions over finite fields in the last years. It is an expanded version of the lectures given by Ph. Michel during the 2016 Arizona Winter School.
(Forum of Math, Sigma, 2020)
Periodic twists of GL3-modular forms, with
, and .
This paper extends the method recently devised by Holowinsky and Nelson (motivated by earlier ideas of Munshi) to prove subconvexity for critical values of L-functions of a fixed GL3-modular form twisted by Dirichlet characters modulo a prime, and proves a general estimate for sums of Hecke-eigenvalues of such a modular form twisted by any q-periodic function with bounded discrete Fourier transform, where the sum has length close to q3/2. This applies in particular to trace functions of Fourier sheaves, and in that respect it is similar to the paper 51 (with Fouvry and Michel); however, it turns out that the problem is somewhat easier, due to the longer length of the sums compared to the modulus (although it is the natural length for applications to subconvexity).
(SMF, Cours spécialisés, 2019, 2021)
An introduction to expander graphs
This is the published version of my lecture notes on expander graphs. A preliminary version is available, as well as a list of corrections (updated Jan. 29, 2020). A second corrected printing appeared in 2021.
(Memoirs of the AMS, 2023)
|The second moment theory of families of L-functions, with , , and and .
Building on our previous works 66 and 68, one can prove asymptotic formulas with power saving for the first and second twisted moments of the family of L-functions of a fixed cusp form twisted by Dirichlet characters of large prime modulus.
It is well-understood that this information, when available for any reasonable family of cusp forms, opens the possibility of many arithmetic applications. In this monograph, we present the general picture first informally, and then derive many of these applications in our particular case.
We also consider some applications that are much more specific to this specific family: (1) the asymptotic computation of the variance of modular symbols of a fixed holomorphic cusp form of level 2 modulo a prime (answering partly a question of Mazur and Rubin), (2) positive proportion of non-vanishing of central values of twisted L-functions with a condition on the discrete Mellin transform of a suitable trace function (this last part relies on the recent work of Katz on Mellin transforms over finite fields).
|Stratification and averaging for exponential sums:
forms with generalized Kloosterman sums,
with and .
This is a follow-up to our paper 68: we refine our methods to obtain non-trivial bounds for bilinear forms with Kloosterman sums with characters in a range of the variables that is identical with the best-known results for much simpler (monomial) exponentials. Analytically, the idea is to use higher Hölder inequalities, but this is not sufficient by itself. The key new geometrical idea is to show that even when the resulting "sum-product" sheaves fail to be irreducible, their decomposition reflects that of the "input" sheaves, except for parameters in a high-codimension subset. This fact is proved by a subtle interplay between étale cohomology in its algebraic and diophantine incarnations. We derive a first application to a first moment of a family of degree 3 L-functions. This preprint was briefly known as "Bilinear forms with Kloosterman sums, II".
(L'enseignement mathématique, 2018)
cyclic codes and the uncertainty principle, with and .
This is a survey of variants of the uncertainty principle over various fields (especially finite fields), and how even a rather weak form of such a result would ensure the existence of sequences of "good" cyclic codes, a long-standing open problem in coding theory. Arithmetically, there are many fascinating connections with the properties of the order of elements of the multiplicative group of a finite field, e.g., Artin's conjecture on primitive roots.
(Proc. Steklov Inst. Math., 2017)
|Some applications of smooth bilinear forms with
Kloosterman sums, with , , and .
This is a complement to our previous paper 66, based on a recent bound of I. Shparlinski and T.P. Zhang for smooth bilinear forms with coefficients Kl2(mn;p).
(in "Exploring the Riemann zeta function", Springer, 2017)
|Bagchi's Theorem for families of automorphic forms.
This is a quick discussion of a GL2 case of Bagchi's Theorem on the functional limit behavior of families of L-functions on the right of the critical line. The argument is adapted from my lecture notes on probabilistic number theory (where the original case of the Riemann zeta function is discussed); see also the summary on my blog.
(Annals of Math., 2017)
|Bilinear forms with Kloosterman sums and applications, with and .
We prove non-trivial estimates for general bilinear forms with coefficients of the type K(mn;p), where K(*;p) are (hyper)Kloosterman sums. This has applications for instance to the average of twisted L-functions of cusp forms, as in paper 66. Indeed, we obtain a power-saving asymptotic formula in that problem.
The heart of the paper is a detailed and subtle study of auxiliary sheaves that we call "sum-product sheaves", which we need to prove are (generically) geometrically irreducible. This requires many deep results from algebraic geometry, going beyond the basic formalism of the Riemann Hypothesis (e.g., to determine precisely the local monodromy of sum-product sheaves at suitable singularities).
(Annales de l'Institut Fourier, 2017)
|On short sums of trace functions, with , , , and .
We consider sums of oscillating functions on intervals in cyclic groups of size close to the square root of the size of the group. We first prove non-trivial estimates for intervals of length slightly larger than this square root (bridging the "Pólya-Vinogradov gap" in some sense) for bounded functions with bounded Fourier transforms. We then prove that the existence of non-trivial estimates for ranges slightly below the square-root bound is stable under the discrete Fourier transform, and we give a number applications related to trace functions over finite fields.
(Remark. This paper contains a considerable strengthening of the basic "sliding sum" bound of reference 58, but not the results on generalized arithmetic progressions there.)
(American J. of Math., 2017)
|On moments of twisted L-functions, with , , and .
We study the average over Dirichlet characters of prime modulus of the product of two central values of twisted L-functions of modular forms. When the two forms are a special Eisenstein series, this is the fourth moment of the central values of the Dirichlet L-functions. Using new bounds for shifted convolution sums (avoiding a dependency on the Ramanujan-Petersson conjecture) and new bounds for bilinear forms in Kloosterman sums, we obtain an asymptotic formula with power saving when one of the two forms is the Eisenstein series; in the second case, this improves (and streamlines) significantly a well-known result of M. Young. We also show how a conjecture on complete two-variable sums over finite fields leads to an asymptotic formula with power saving for two cusp forms.
(Remark. The conjecture mentioned was proved by Ph. Michel, W. Sawin and myself in the paper 68, hence the asymptotic formula for two cusp forms is now unconditional.)
(Compositio Math., 2016)
|Kloosterman paths and the shape of exponential sums, with .
We study the statistics of the shape of the graphs joining successive partial sums of Kloosterman sums (see the Kloostermania page), as well as of other types of exponential sums. We prove convergence in the space of continuous functions for a suitable family, with limit given by a specific remarkable random Fourier series. This result turns out to depend on Katz's determination of the monodromy groups of Kloosterman sheaves (and the Riemann Hypothesis over finite fields), and on estimates for averages of short sums at the edge of the Pólya-Vinogradov range. Using some probabilistic results of Talagrand and Montgomery-Smith in particular, we deduce bounds on the tails of the maximum of the normalized partial sums.
An introduction to the representation theory of groups
This book is based on my introductory lecture notes on representation theory, and presents the fundamental concepts of representation theory for groups, as well as some applications. I have started collecting some corrections and additions.
(Phil. Trans. Royal Soc. A, 2015)
|A study in sums of products, with and .
This is to a large extent a survey of the applications of the ideas of Katz surrounding the Goursat-Kolchin-Ribet criterion for monodromy groups of direct sums of sheaves to the estimation of sums over finite fields where the summands are products of trace functions of suitable sheaves. Such sums arise frequently in analytic number theory, and we provide a convenient reference for general statements as well as for important special cases. We illustrate the method with a number of applications, old and new. A shorter version appears in the proceedings of the Royal Society Meeting Number fields and function fields: coalescences, contrasts and emerging applications.
(Bourbaki seminar, 2014)
between prime numbers and prime numbers in arithmetic
progressions, after Y. Zhang and J. Maynard
This is the written English version of my lecture at the Bourbaki seminar in March 2014 concerning the work of Y. Zhang and J. Maynard on gaps between primes. A French translation is in preparation.
(Colloquium De Giorgi, 2015)
|Trace functions over finite fields and their applications, with and .
This is a survey of (most) of our papers on trace functions, with an emphasis on 51, 55 and 57. It is a written and slightly expanded version of a lecture I gave in early 2013 at the Colloquio de Giorgi of the Scuola Normale Superiore di Pisa, and appears in the series of proceedings volumes.
|Fourier coefficients of GL(N) automorphic forms in arithmetic progressions, with .
This paper generalizes the results of my previous paper with É. Fouvry, S. Ganguly and Ph. Michel (number 56 below) to all cusp forms on GL(n) for all N≥ 3, and to the N-ary divisor function dN, obtaining a Gaussian distribution result for Fourier coefficients in arithmetic progressions in suitable ranges. This is therefore a new case of an analytic property that is now known for automorphic forms on all linear groups GL(N)! From this point of view, one can note that, besides the Rankin-Selberg convolution and bounds towards the Ramanujan-Petersson conjecture, which occur in the proofs of other "universal" results, we require deep equidistribution statements for products of hyper-Kloosterman sums.
|On the conductor of cohomological transforms, with and .
We show that the conductor of ℓ-adic sheaves on the affine line defined by "cohomological transforms" with a kernel given by an Artin-Schreier sheaf in two variables is bounded in terms of the conductor of the input sheaf and the degree of the numerator and denominator of the rational function defining the kernel. In particular, this applies to the Fourier transform in one variable, and recovers in that case a result we used in our previous works and which we had deduced from the deep local Fourier transform theory of Laumon.
W. Sawin has developed a much more general theory (now published in the paper 82, co-written with A. Forey, J. Fresán and myself), so this paper is mostly of historical interest. It can however be useful as an illustration of elementary cohomological techniques, going a few steps beyond the formalism of the Riemann Hypothesis.
|The sliding sum method for short exponential sums, with and .
We develop a very simple idea to obtain non-trivial estimates for sums of suitable functions over intervals in a finite abelian group (or more generally, over proper generalized arithmetic progressions) when the size of the summation set is just within the gap between the square-root of the size of the group and the range where the Pólya-Vinogradov completing method becomes relevant. The condition on the function is related to the size of its additive correlation sums, and we show that trace functions satisfy this condition, except when it is obviously false. We then derive some applications to equidistribution in the same ranges.
(Remark. The basic inequality has been considerably strengthened in reference 67, jointly with CS. Raju, J. Rivat and K. Soundararajan; some results on generalized arithmetic progressions are however not considered in that later paper.)
|On the exponent of distribution of the ternary divisor function, with and .
This paper is an illustration of the power and flexibility of the results in our first two papers (number 51 and 55 below) concerning trace functions over finite fields and their uses in analytic number theory. We give a much more streamlined proof of the result of Friedlander and Iwaniec (later improved by Heath-Brown) that the exponent of distribution of the ternary divisor function in arithmetic progressions (to prime moduli at least) is strictly larger than 1/2, and in fact obtain a value for this exponent which also improves the one of Heath-Brown. Combining this with "Kloostermania", we also improve the exponent of distribution on average over (prime) moduli for a fixed residue class.
(Commentarii Math. Helv., 2014)
|Gaussian distribution for the divisor function and Hecke eigenvalues in arithmetic progressions, with , and .
We prove that the error term of the divisor function in arithmetic progressions modulo a prime number p has a Gaussian limiting distribution as a function of the residue class, in a suitable range. We also show a similar result for Hecke eigenvalues of classical primitive holomorphic cusp forms of level 1, and determine when the error term for the residue classes a and γ(a) are asymptotically independent, for a projective linear transformation γ modulo p. The proof uses the moment method and involves the Voronoi summation formula and a crucial use of the Riemann Hypothesis over finite fields, in the form of a result of independence of twists of Kloosterman sheaves.
(Duke Math. Journal, 2014)
|Algebraic trace functions over the primes, with and .
We prove that sums over primes of trace functions of middle-extension sheaves on A1/Fp are small, when the length is close enough to p in logarithmic scale, and a similar statement for correlation of trace functions with the Möbius function. The bound is valid for any trace function which is not proportional to the product of an additive and a multiplicative character. We then prove a number of applications (estimates for sums of multiplicative characters of a rather general polynomial over the primes, existence of a positive density of "large" Kloosterman sums with moduli having two prime factors, among others.)
(Math. Proc. Cambridge Phil. Soc., 2013)
|An inverse theorem for Gowers norms of trace functions over Fp, with and .
We prove that the Gowers uniformity norms of functions over a prime field which arise as trace functions of a middle-extension sheaf on the affine line are asymptotically "as small as possible", except if the sheaf contains Artin-Schreier sheaves of polynomials. This is a strong form of inverse theorem for these functions with algebraic structure.
(Math. Res. Letters, 2013)
|Counting sheaves using spherical codes, with and .
Using a result on the size of spherical codes in a suitable range, we obtain here a polynomial-type bound (in terms of the size of the finite field) for the number of lisse sheaves on an algebraic curve over a finite field which have bounded conductor (i.e., bounded rank and ramification). As an application, we show that a function defined on a finite field can not usually be expressed as a short linear combination of trace functions of sheaves with small complexity.
(MSRI publications, 2014)
|Sieve in discrete groups, especially sparse
This is the written version of the mini-course I gave during the "Hot Topics" workshop at MSRI on "Thin groups and super-strong-approximation" that I co-organized in February 2012. It has a certain intersection with my Bourbaki seminar talk on sieve and expanders (see paper number 47 below) but I have tried to make this survey complementary of the previous one. In particular there is more discussion of the recent versions and applications of the very general large sieve inequality by Lubotzky, Meiri and Rosenzweig. There is also a discussion of an Erdös-Kac Theorem in the context of the affine sieve.
|Algebraic twists of modular forms and Hecke orbits, with and .
We consider correlations of Fourier coefficients of classical modular forms with functions "of algebraic origin", defined as trace functions of suitable sheaves on A1/Fp. We are able to show a strong form of asymptotic orthogonality. Analytically, the main tools are the amplification method and the Kuznetsov formula, and geometrically we depend crucially on the full force of the Riemann Hypothesis over finite fields, and on the properties of the Fourier transform for sheaves, as defined by Deligne and studied by Katz, Laumon and others. This leads to applications concerning the distribution of "twisted" Hecke orbits, or discrete horocycles, on modular curves.
|Explicit growth and expansion for SL2
This contains a streamlined self-contained account of fully explicit versions of the growth theorem of Helfgott and of the Bourgain-Gamburd expansion theorem for the case of SL2(Fp).
Warning! The current version (updated on July 2, 2012) contains many changes and corrections, compared with the first draft, some of them quite important, and including all (!) numerical constants; many thanks to the remarkable work of the referee on this paper.
(Pub. Math. Besançon, 2013)
|Families of cusp forms
This survey describes some ideas towards a definition (or some understanding...) of the notion of a family of cusp forms. This is influenced by similar ideas of P. Sarnak, and related to earlier work of Cogdell-Michel, as well as to the "recipe" of Conrey, Farmer, Keating, Rubinstein and Snaith for the prediction of moments of L-functions on the critical line.
|Mod-φ convergence, with and .
We discuss a generalization of the notion of mod-Gaussian convergence, which we call "mod-φ convergence". This applies to much more general model distributions than the Gaussian ones, and we show how, in this context, very simple conditions lead to a local limit theorem when mod-φ convergence holds. We derive many applications, from the winding number of planar brownian motion to models of random squarefree integers, and show that this framework suggests a strong form of the conjecture that the values of the Riemann zeta function on the critical line are dense in the complex plane. This paper is dedicated to the memory of Marc Yor.
|Sieve in expansion and Crible en expansion
This is a survey paper on the recent developments of sieve involving expander graphs, notably the "sieve in orbits" of Bourgain, Gamburd and Sarnak, with a few quick asides concerning geometric applications (e.g., to Dunfield-Thurston random 3-manifolds). It is the written counterpart of a lecture at the Séminaire Bourbaki. The French version (translation) is essentially identical with the English one.
(Compositio Math., 2012)
Local spectral equidistribution for Siegel modular forms and applications, with and .
This is a first exploration, in the case of a group of rank 2 which is not a general linear group, of the philosophy that says (roughly) that good "families" of cusp forms should have the property that, for any fixed prime p, their p-component (in the sense of automorphic representations) should behave well, and in particular become equidistributed in a suitable space of Satake parameters, with respect to some measure, possibly depending on the prime (there is a semi-philosophical discussion in this talk that I gave at R. Bruggeman's 65th birthday conference in 2009). We study this local distribution question for Siegel cusp forms of genus 2, where the parameter growing to infinity is the weight. Our averages are performed subject to a specific weighting which allows us to apply results concerning Bessel models and a variant of Petersson's formula. We obtain for this family a quantitative local equidistribution result, and derive a number of consequences. In particular, we show that the computation of the density of low-lying zeros of the degree 4 spinor L-functions (for restricted test functions) gives global evidence for a well-known conjecture of Böcherer concerning the arithmetic nature of Fourier coefficients of Siegel cusp forms, which involves central values of L-functions.. (Note that qualitative equidistribution statements can be proved more easily using the approximate orthogonality of Fourier coefficients of Poincaré series proved in our earlier paper, number 43 below).
(Israel J. Math., 2012)
Splitting fields of characteristic polynomials of random elements in arithmetic groups, with and .
We discuss rather systematically the principle, implicit in our earlier paper 32 and in other works, that for a "random" element in an arithmetic subgroup of a (split, say) reductive algebraic group over a number field, the splitting field of the characteristic polynomial, computed using the any faitful representation, has Galois group isomorphic to the Weyl group of the underlying algebraic group. Besides tools such as the large sieve, which we had already used, we introduce some probabilistic ideas (large deviation estimates for finite Markov chains) and the general case involves a more precise understanding of the way Frobenius conjugacy classes are computed for such splitting fields (which is related to a map between regular elements of a finite group of Lie type and conjugacy classes in the Weyl group which had been considered earlier by Carter and Fulman for other purposes; we show in particular that the values of this map are equidistributed).
(Duke Mathematical Journal, 2012)
Expander graphs, gonality and variation of Galois representations, with and .
We show that strong finiteness theorems concerning variation of Galois representations (e.g., with respect to maximal Galois action on torsion points of one-parameter families of abelian varieties) over number fields can be obtained using expansion properties of Cayley graphs attached to a family of coverings of the base curve, using lower-bounds for the gonality in such families that arise by combining a result of Li and Yau with the comparision principle of Brooks and Burger relating the first non-zero eigenvalue of the Riemannian and combinatorial Laplace operators for hyperbolic surfaces. Although expander families are natural targets for our methods, a weaker notion of expansion (which we call esperantism) is also sufficient. This surprising connection leads also to many intriguing questions.
This paper is in some sense a continuation of the paper 34 below (also joint with ); additional discussion can be bound in this blog post of J. Ellenberg, and this other one.
A note on Fourier coefficients of Poincaré series, with and .
We give a short and "soft" proof of the qualitative approximate orthogonality of Fourier coefficients of Poincaré series for classical and Siegel modular forms, in the limit when the weight goes to infinity.
(Journal of the LMS, 2012)
Mod-Gaussian convergence and the value distribution of ζ(1/2+it) and related quantities, with .
A difficult question in the theory of the Riemann zeta function (first raised by Ramachandra) asks whether the set of values ζ(1/2+it), t real, is dense in the complex plane. We provide evidence for an affirmative answer (in a strong quantitative form) by relating the problem to mod-Gaussian convergence (as defined in our paper 35 with J. Jacod) and the moment conjectures for the zeta function. This remains conditional, but we obtain results of independent interest concerning values of characteristic polynomials of random matrices in compact classical groups, and from this derive similar results for values of L-functions over finite fields -- e.g. the central values of L-functions of Dirichlet characters over Fp[T] form a dense subset of the complex plane, where p runs over primes. (Version updated May 21, 2010).
Mod-discrete expansions, with and .
This is a systematic exploration of the probabilistic consequences of conditions similar to mod-Poisson convergence, in terms of approximation (in Kolmogorov or total variation sense) of discrete random variables by Poisson variables, Poisson-Charlier expansions, or similar constructions. In particular, this provides approximations in total variation norm for the number of prime factors of integers up to N, with error that can be an arbitrary inverse power of log log N. This is also partly motivated by earlier results of Hwang (based on probability generating functions).
(Math. Proc. Cambridge Phil. Soc., 2010)
On modular signs, with , and .
We consider in a fairly systematic way the properties of the sequence of signs of Hecke eigenvalues of classical primitive modular forms, obtaining results concerning the first sign change (improving strongly a result of Iwaniec, Kohnen and Sengupta), "multiplicity one" theorems for the sequence of signs (with small set of exceptions), and statistical upper and lower bounds for the number of forms with a given set of signs for the first few primes.
(Milan Journal of Mathematics, 2010)
Some aspects and applications of the Riemann Hypothesis over finite fields.
This is a survey of some of the applications of the Riemann Hypothesis over finite fields, in the realm of analytic number theory, with a particular emphasis on presenting Deligne's Equidistribution Theorem and some of its applications. This is the written version of my lecture at the Verbania Conference on the 150th anniversary of the Riemann Hypothesis.
Mod-Poisson convergence in probability and number
theory, with .
Building on our previous work with J. Jacod (paper 35), we consider in more detail the "mod-Poisson" convergence of random variables. Besides pointing out various instances in probability, we emphasize its presence in the distribution of the number of prime and irreducible factors of integers and polynomials over finite fields. In that second case, we show that the result amounts to a zero-dimensional case of the Katz-Sarnak philosophy in the limit of large conductor. See this post for an explanation of the analogy involved.
(Archiv der Mathematik, 2010)
Amplification arguments for large sieve inequalities.
This paper combines an earlier note giving an alternate proof of the arithmetic large sieve inequality, based on amplification ideas, and parts of the note on "modular signs", linked together by means of another result on the statistical paucity of primitive modular forms for which the first few Hecke eigenvalues have the same sign; this last result is deduced from a large-sieve type inequality which has the feature of (partly) unifying the classical large sieve and the spectral large sieve inequalities for Fourier coefficients of cusp forms, and the proof depends on using the amplification method.
(Experimental Mathematics, 2012)
The Chebotarev invariant of a finite group, with
We consider invariants of a finite group G which are related to the number of random (independent, uniformly distributed) conjugacy classes which are required to generate it; this is related, intuitively at least, to the number of primes required to determine the Galois group of an integral polynomial when computing Frobenius conjugacy classes (though the precise link between the model considered here and the actual arithmetic situations requires some care). This paper gives purely group-theoretic expressions for the invariants, and uses them to investigate their values, both theoretically for certain families of groups which are easily accessible, and numerically using Magma. For a more informal description, see this blog post.
(Forum Mathematicum, 2011)
Mod-Gaussian convergence: new limit theorems in probability and number theory, with and .
We show that certain theorems and conjectures in Random Matrix Theory can be interpreted as instances of a new type of limit theorems in probability theory, which in a sense "refine" standard results of convergence to the normal distribution. A similar phenomenon, with Poisson distribution instead, turns out to be present in the classical theorem of Erdös and Kác on the distribution of the number of prime factors of integers. Also arXiv:0807.4739.
(Journal of the LMS, 2009)
Non-simple abelian varieties in a family: geometric and analytic approaches, with , and .
We look at a single question using two quite different approaches, and obtain complementary answers... The question is to "count" how many fibers (in a number field) of a generically simple abelian variety remain simple, and the answer is always that not many do. The methods of arithmetic geometry (around monodromy groups) lead, using the theorem of Faltings, to the result that there are only finitely many exceptions, but not to quantitative information (in terms of the family). Sieve techniques (based on mixing the sieve for Frobenius and a version of Gallagher's "larger" sieve) give an estimate which grows to infinity with the permitted height of the exceptions... but which is effective and fairly well-controlled. So depending on the point of view, either may be more useful! A short discussion is in this blog post of J. Ellenberg. (See also a paper of Y. Zarhin, who finds many examples where the set of exceptions is actually empty). Also arXiv:0804.2166.
(Acta Arithmetica, 2011)
Averages of Euler products, distribution of singular series
and the ubiquity of Poisson distribution
This paper extends a well-known result of Gallagher on the average value of the singular series for the prime k-tuple conjecture in various directions. For instance, the (conditional) applications to the distribution of primes in intervals of "fair" length are extended to the distribution of any prime patterns in such intervals: if uniform versions of the Bateman-Horn conjecture hold, there will be convergence to a Poisson distribution with the appropriate mean. In addition, the proofs are rather more transparent and intuitive. Also arXiv:0805.4682.
(Cambridge University Press, 2011)
A survey of algebraic exponential sums and some applications
This is an expanded version of a talk I gave at a (wonderful!) workshop on Motivic Integration and its interactions with model theory and non-archimedean geometry, organized in 2008 by R. Cluckers, A. Macintyre, J. Nicaise and J. Sebag; it appears as a chapter in the proceedings volume. This survey of exponential sums over finite fields overlaps a bit -- but surprisingly little -- with the other one I wrote later for the Verbania conference (item 39); the main originality is the discussion of exponential sums over definable sets over finite fields, following the paper 26. (Note that this paper is numbered 32bis because I had forgotten to include it on the web page at the right time...)
An explicit integral polynomial whose splitting field has
Galois group W(E8),
By exploiting ideas about the "generic" behavior of the splitting field of characteristic polynomials of elements of linear algebraic groups, we construct an explicit polynomial P of degree 240 with integral coefficients such that the Galois group of its splitting field is the Weyl group of the exceptional group of type E8. (Apparently, the first such polynomial to be written down, though the existence has been known for a long time; the largest coefficients of P have about a hundred digits). This paper is dedicated to H. Cohen and appears in the special volume of the Journal de Théorie des Nombres de Bordeaux published in his honor. Also arXiv:0801.1733.
The large sieve,
monodromy and zeta functions of algebraic curves, II:
Independence of the zeros
Using the sieve for Frobenius, we show that there are "usually" no multiplicative or Q-linear relations between zeros of L-functions of algebraic curves over finite fields, which is an analogue of well-known linear independence conjectures for zeros of Dirichlet L-functions and has consequences, for instance, for determining the distribution of the difference between the number of points of two curves over extensions of the base field (this is an analogue of the Chebychev bias... except that here there is typically no bias). In the case (of most interest) of multiplicative relations, we outline an alternative approach, based on the theory of Frobenius tori of Serre (this was suggested by N. Katz). Also arXiv:0807.2118.
(Cambridge University Press, 2008)
The large sieve and its applications
This is the expanded version of the preprint "The principle of the large sieve" (see arXiv:math.NT/0610021). The general sieve framework developed in this book is also described in a guest post on T. Tao's blog. Here are the current corrections, and a short addition that explains how to derive the arithmetic large sieve inequality from the "dual" inequality in the general setting of the book.
(Princeton University Press, 2007)
Mathematics for physics... and for physicists!
(English translation of «Mathématiques pour la physique... et les physiciens !», by Walter Appel, published by Éditions H&K)
entre nombres premiers (d'après Goldston, Pintz,
Séminaire Bourbaki, Exposé 959 (2006).
|Poincaré et la
théorie analytique des nombres
(in «L'Héritage scientifique d'Henri Poincaré», edited by É. Charpentier, É. Ghys et A. Lesne); an English translation is published by the AMS, and here is my own version.
(Israel J. Math., 2007)
sums over definable subsets of finite fields
Also arXiv: math.NT/0504316
(Archiv der Mathematik, 2007)
|Équirépartition adélique de mesures algébriques dans un groupe résoluble et sommes de Kloosterman
(Int. J. of Number Theory, 2006)
the rank of quadratic twists of elliptic curves over function
Also arXiv: math.NT/0503732
(American Math. Monthly, 2006)
polynomials and Brownian motion
Also see Problem 11155 in the May, 2005, issue of the Monthly (solution by T. Rivoal). Proposition 6.1 on zeros of random Bernstein polynomials is false! This was pointed out by Johannes Ruf; I was misremembering some properties of zeros of Brownian motion...
large sieve, monodromy and zeta functions of curves
Also arXiv: math.NT/0503714. Here are a few small corrections (also contained in the forthcoming book "The large sieve and its applications").
(Journal of the L.M.S, 2006)
numbers generated by other Weil numbers and torsion fields of abelian
Also arXiv: math.NT/0504042. Note that Remark 3.9 (which is Remark 6 in Section 3 in the printed version) refers to a question (on the behavior of the splitting type of abelian varieties when reducing modulo primes) which was supposedly mentioned in an earlier paper (number 12), but in fact I had commented-out this remark during final corrections.
(Archiv der Mathematik, 2005)
|Variations of recognition problems for modular forms
(American Math. Soc., 2004)
See also the Current list of corrections and (forthcoming) Supplementary material. There is also a Russian translation (published by MCCME, 2014).
(Cambridge University Press, 2007)
Elliptic curves, rank in families and random matrices
(Typed version of a survey lecture given at the Newton Institute Workshop and random matrices and L-functions in July 2004, together with a survey lecture on the variation of the rank of elliptic curves in families at the AIM/Princeton University workshop on the Birch and Swinnerton-Dyer Conjecture in November 2003)
Small gaps in coefficients of L-functions and B-free
numbers in short intervals,
Also arXiv: math.NT/0507001. Updated in November 2019, to correct a minor mistake in Lemma 2.2 pointed out by Narasimha Kumar; this does not affect the main results.
(American Math. Monthly, 2004)
On the reducibility of arctangents of integers
See also Sequence A002312 in the Encyclopedia of Integer Sequences. (This is a short application of the result of Duke, Friedlander and Iwaniec proved in the graduate course below).
(SMF Cours Spécialisés, 2004)
Un cours de théorie analytique des nombres
(DEA/graduate course in Bordeaux, 2001-2002).
Automorphic forms, L-functions and number theory: 3
See some corrections to the published version.
|Dependency on the group in automorphic Sobolev inequalities
(Manuscripta Math., 2003)
Local-Global Applications of Kummer Theory
Note that Proposition 6.11 is not in the published version
(Pacific J. Math., 2002).
|Zeros of families of automorphic L-functions close to 1, with
(J. Ramanujan Math. Society, 2006)
problems for elliptic curves
Also arXiv: math.NT/0510197
(Duke Math. J., 2002)
|Rankin-Selberg L-functions in the level aspect, with and
(Invent. math., 2000)
|Mollification of the fourth moment of automorphic L-functions and arithmetic applications, with and
(Israel J. Math., 2000)
|Explicit upper bound for the (analytic) rank of J0(q), with
(Manuscripta Math., 2001)
|Deux théorèmes de non-annulation de valeurs spéciales de fonctions L, with
(Duke Math. J., 2001)
Vérification de l'hypothèse Hp(chi) pour p grand,
Appendix to a paper by L. Merel
|Non-vanishing of high derivatives of automorphic L-functions at the center of the critical strip, with and
(Acta Arithetica, 2000)
|A lower bound for the rank of J0(q), with
(Duke Math. J., 2000)
|The analytic rank of J0(q) and zeros of automorphic L-functions, with
(Invent. math., 2000)
|A problem of Linnik for elliptic curves and mean value estimates for automorphic representations, with and
Last update 10.1.2024 by E. Kowalski