The ubiquity of
surjective reduction in random groups
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.
bounds for orthonormal basis elements in Hilbert spaces
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 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.
A funny identity
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
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).
certain bad behavior of Fourier coefficients of modular forms
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
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
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
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
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 are incorporated in the preprint The principle of the large sieve and in the forthcoming book The large sieve and its applications.
symplectic monodromy: a theorem of C. Hall
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.
two-dimensional larger sieve
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).
de deux carrés successifs qui sont des carrés
This explains that there are infinitely many integers such that n2+(n+1)2 is a square. The values of n form sequence A001652 of the Online Encyclopedia of Integer Sequences (in particular the problem is fairly classical, but I didn't know this until searching for the sequence...)
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 (and really unconditional) 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).
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.
global root number for J0(q)
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.
curves in the plane
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.
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
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.
d'Atkin-Lehner pour les formes paraboliques de Maass,
représentations automorphes de GL(2)
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...
fonction de Beurling et la grand crible. Applications.
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...