Papers, books, translations, surveys
| 50 |
Research paper
(preprint, 2012)
|
Explicit growth and expansion for SL2
This contains a streamlined self-contained account of the explicit versions of the growth theorem of Helfgott and of the Bourgain-Gamburd expansion theorem for the case of SL2(Fp) that are found in Chapter 4 of my lecture notes on expander graphs. The treatment is also more complete, with explicit forms of what "p large enough" means in the expansion theorem. |
| 49 |
Survey
(Informal, 2011)
|
Families of cusp forms
In this survey paper (which is not currently intended for publication), we describe 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. |
| 48 |
Research paper
(Preprint, 2011)
|
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. |
| 47 |
Survey
(Association Bourbaki, 2010)
|
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. |
| 46 | Research paper (to appear, Compositio Math.)
|
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). |
| 45 | Research paper (to appear, Israel J. Math.)
|
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). |
| 44 | Research paper (to appear, Duke Mathematical Journal)
|
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. |
| 43 | Research paper (Mathematika, 2011)
|
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. |
| 42 | Research paper (to appear, Journal of the LMS)
|
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). |
| 41 | Research paper (preprint)
|
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). |
| 40 | Research paper (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. |
| 39 | Survey (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. |
| 38 | Research paper
(IMRN, 2010)
|
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. |
| 37 | Research
paper (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. |
| 36 | Research paper (to appear, Experimental Mathematics)
|
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, as will be considered in another paper). 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. |
| 35 | Research paper (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. |
| 34 | Research paper (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. |
| 33 | Research paper (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. |
| 32bis | Survey (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 will appear 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...) |
| 32 | Research paper |
An explicit integral polynomial whose splitting field has
Galois group W(E8),
with
and .
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. |
| 31 | Research paper (IMRN, 2008)
|
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. |
| 30 | Book
(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. |
| 29 | Book translation
(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) |
| 28 |
Survey
(Association Bourbaki,
Astérisque,
2007)
|
Écarts successifs
entre nombres premiers (d'après Goldston, Pintz,
Yıldırım) Séminaire Bourbaki, Exposé 959 (2006). |
| 27 | Popular survey
(Belin-Échelles,
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. |
| 26 | Research paper
(Israel
J. Math., 2007)
|
Exponential
sums over definable subsets of finite fields
Also arXiv: math.NT/0504316 |
| 25 | Research paper
(Archiv
der Mathematik, 2007)
|
Équirépartition adélique de mesures algébriques dans un groupe résoluble et sommes de Kloosterman |
| 24 | Research paper
(Int. J. of
Number Theory, 2006)
|
On
the rank of quadratic twists of elliptic curves over function
fields
Also arXiv: math.NT/0503732 |
| 23 | Research paper
(American
Math. Monthly, 2006)
|
Bernstein
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... |
| 22 | Research paper
(Crelle,
2006)
|
The
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"). |
| 21 | Research paper
(Journal
of the L.M.S, 2006)
|
Weil
numbers generated by other Weil numbers and torsion fields of abelian
varieties
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. |
| 20 | Research paper
(Archiv
der Mathematik, 2005)
|
Variations of recognition problems for modular forms |
| 19 | Book
(American
Math. Soc., 2004)
|
Analytic
Number Theory,
with
See also the Current list of corrections and (forthcoming) Supplementary material. |
| 18 | Survey
(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) |
| 17 | Research paper |
Small gaps in coefficients of L-functions and B-free
numbers in short intervals,
with
and
Also arXiv: math.NT/0507001 |
| 16 | Research note
(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). |
| 15 | Book
(SMF
Cours Spécialisés, 2004)
|
Un cours de théorie analytique des nombres
(DEA/graduate course in Bordeaux, 2001-2002). |
| 14 | Book chapters
(Birkhäuser,
2003)
|
Automorphic forms, L-functions and number theory: 3
lectures
See some corrections to the published version. |
| 13 | Research paper (to appear) | Dependency on the group in automorphic Sobolev inequalities |
| 12 | Research paper
(Manuscripta
Math., 2003)
|
Some
Local-Global Applications of Kummer Theory
Note that Proposition 6.11 is not in the published version |
| 11 | Research paper
(Pacific J. Math.,
2002).
|
Zeros of families of automorphic L-functions close to 1, with |
| 10 | Research paper
(J. Ramanujan
Math. Society, 2006)
|
Analytic
problems for elliptic curves
Also arXiv: math.NT/0510197 |
| 9 | Research paper
(Duke Math. J.,
2002)
|
Rankin-Selberg L-functions in the level aspect, with and |
| 8 | Research paper
(Invent.
math., 2000)
|
Mollification of the fourth moment of automorphic L-functions and arithmetic applications, with and |
| 7 | Published
(Israel J.
Math., 2000)
|
Explicit upper bound for the (analytic) rank of J0(q), with |
| 6 | Research paper
(Manuscripta
Math., 2001)
|
Deux théorèmes de non-annulation de valeurs spéciales de fonctions L, with |
| 5 | Appendix
(Duke Math. J.,
2001)
|
Vérification
de l'hypothèse Hp(chi)
pour p grand,
with
Appendix to a paper by L. Merel |
| 4 | Research paper
(Crelle,
2000)
|
Non-vanishing of high derivatives of automorphic L-functions at the center of the critical strip, with and |
| 3 | Research paper
(Acta
Arith., 2000)
|
A lower bound for the rank of J0(q), with |
| 2 | Research paper
(Duke Math. J.,
2000)
|
The analytic rank of J0(q) and zeros of automorphic L-functions, with |
| 1 | Research paper
(Invent.
math., 2000) |
A problem of Linnik for elliptic curves and mean value estimates for automorphic representations, with and |
Last update 1.1.2012 by E. Kowalski