Prepublications

• Sommes de carrés, a lesson of Philippe Michel at "Leçons de Mathématiques et d'Informatique d'aujourd'hui", to be published in Cassini, Collection Le Sel et le Fer (2019).

• A new bound for the sup-norm of automorphic forms on non-compact modular curves in the level aspect, with Harald Helfgott (2019).
  
  We prove that the sup-norm of a L²-normalised Hecke Maaβ cusp form of level N is bounded by N^-1/20.

• An explicit density estimate, with at least 39 coauthors, submitted (2019).
  
  The number of co-authors is unusual in the field of Number Theory (and in Pure Mathematics too) but it reflects a political decision against a certain policy of systematical bibliometrical evaluation. Indeed bibliometrical informations are starting to be used systematically. For instance, a three pages long paper with 582 co-authors has thus served as a basis to distribute to one of its author a huge part of the department budget in Lyon. In front of the uproar, this university has finally given up on this decision, but only for this year. Several similar steps are being taken, and in other European countries as well. This paper is not a trick and is completely correct. It can be seen as a natural mutation to get adapted to this everchanging world. In this paper, we prove an explicit upper bound for the number of zeros for L-functions, whose imaginary part is less than T and real part larger than σ>1/2.
  
  
  
  

Publications

• Kloosterman paths of prime powers moduli, II, with Emmanuel Royer and Igor Shparlinski, published in Bulletin de la Société mathématique de France, vol. 148, 173--188 (2020).
  
  G. Ricotta and E. Royer (2018) have recently proved that the polygonal paths joining the partial sums of the normalized classical Kloosterman sums S(a,b;pⁿ)/p^n/2 converge in law in the Banach space of complex-valued continuous function on [0,1] to an explicit random Fourier series as (a,b) varies over (Z/pⁿZ)*×(Z/pⁿZ)*, p tends to infinity among the odd prime numbers and n larger than 2 is a fixed integer. This is the analogue of the result obtained by E. Kowalski and W. Sawin (2016) in the prime moduli case. The purpose of this work is to prove a convergence law in this Banach space as only a varies over (Z/pⁿZ)*, p tends to infinity among the odd prime numbers and n larger than 31 is a fixed integer.
  
  
  

• Distribution of short sums of classical Kloosterman sums of prime powers moduli, published in Annales Mathématiques Blaise Pascal, vol. 26, no. 1, 101--117 (2019).
  
  Corentin Perret-Gentil proved, under some very general conditions, that short sums of l-adic trace functions over finite fields of varying center converges in law to a Gaussian random variable or vector. The main inputs are Deligne's equidistribution theorem and Katz' works. In particular, this applies to 2-dimensional Kloosterman sums Kl_2,Fq studied by Katz when the field F_q gets large. This article considers the case of short sums of normalized classical Kloosterman sums of prime powers moduli pⁿ, as p tends to infinity among the prime numbers and n larger than 2 is a fixed integer. A convergence in law towards a real-valued standard Gaussian random variable is proved under some very natural conditions.
  
  
  

• On the sup-norm of SL(3) Hecke-Maass cusp form, with Roman Holowinsky, Kevin Nowland and Emmanuel Royer, published in Publications Mathématiques de Besançon, no.2 , 53—80 (2019).
  
  This work contains a proof of a non-trivial explicit quantitative bound in the eigenvalue aspect for the sup-norm of a SL(3,Z) Hecke-Maass cusp form restricted to a compact set.

• Kloosterman paths of prime powers moduli, with Emmanuel Royer, published in Commentarii Mathematici Helvetici, vol. 93, no. 3, 493—532 (2018).
  
  Emmanuel Kowalski and William Sawin proved, using a deep independence result of Kloosterman sheaves, that the polygonal paths joining the partial sums of the normalized classical Kloosterman sums of S(a,b;p)/p^1/2 converge in the sense of finite distributions to a specific random Fourier series, as a varies over (Z/pZ)*, b is fixed in (Z/pZ)* and p tends to infinity among the odd prime numbers. This article considers the case of S(a,b;pⁿ)/p^n/2, as a varies over (Z/pⁿZ)*, b is fixed in (Z/pⁿZ)*, p tends to infinity among the odd prime numbers and n larger than 2 is a fixed integer. A convergence in law in the Banach space of complex-valued continuous function on [0,1] is also established, as (a,b) varies over (Z/pⁿZ)×(Z/pⁿZ)*, p tends to infinity among the odd prime numbers and n larger than 2 is a fixed integer. This is the analogue of the result obtained by Emmanuel Kowalski and William Sawin in the prime moduli case.

• The amplification method in the context of GL(n) automorphic forms, published in Functiones et Approximatio Commentarii Mathematici, vol. 54, no. 2, 195—226 (2016).
  
  Silverman-Venkatesh and Blomer-Maga proved the existence of a so-called higher rank amplifier and Holowinsky-Ricotta-Royer described an explicit version of a GL(3) amplifier. This article provides, for n larger than 4, a totally explicit GL(n) amplifier and gives all the results required to use it effectively.

• The amplification method in the GL(3) Hecke algebra, with Roman Holowinsky and Emmanuel Royer, published in Publications Mathématiques de Besançon, 13—40 (2015).
  
  This article contains all of the technical ingredients required to implement an effective, explicit and unconditional amplifier in the context of GL(3) automorphic forms. In particular, several coset decomposition computations in the GL(3) Hecke algebra are explicitly done.

• Fourier coefficients of GL(N) automorphic forms in arithmetic progressions, with Emmanuel Kowalski, published in Geometric and Functional Analysis, vol. 24, no. 4, 1229—1297 (2014).
  
  This paper generalizes the results of Fouvry, Ganguly, Kowalski and Michel to all cusp forms on GL(n) for all N larger than 3, and to the N-ary divisor function, 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

• Mean-periodicity and zeta functions, with Ivan Fesenko and Masatoshi Suzuki, published in Annales de l'Institut Fourier, Vol. 62, no. 5, 1819—1887 (2012).
  
  The general admitted expectation is that the right objects parametrizing L-functions are automorphic representations. In this joint work with Ivan Fesenko and Masatoshi Suzuki, it is suggested that the right objects parametrizing Hasse zeta functions of arithmetic schemes are mean-periodic functions over the real line, which have at most polynomial growth. Such Hasse zeta functions are conjecturally ratios of L-functions. As a consequence, the traditional way to prove the expected analytic properties of such Hasse zeta functions is to prove automorphic properties of each of the conjectural L-factors, which is not entirely satisfactory. It is shown in this work that establishing the expected analytic properties of these zeta functions boils down to proving the mean-periodicity of some explicit functions on the real line.The case of regular models of zeta functions of elliptic curves is carefully analysed in this paper.

• Statistics for low-lying zeros of symmetric power L-functions in the level aspect, with Emmanuel Royer, published in Forum Mathematicum 23, 969—1028 (2011).
  
  In this paper, we compute the one-level density and the two-level density for low-lying zeros of some families of symmetric power L-functions in the level aspect. These families are built according to the value of the sign of the functional equation. As a consequence, we completely determine the symmetry types of these families. The main technical ingredients are some large sieve inequalities for Kloosterman sums and Riemann-von Mangoldt’s explicit formula.We also compute the moments of the one-level density and produces a new instance for Hughes-Rudnick’s mock Gaussian behaviour. This result relies on heavy combinatorial arguments.

• Lower order terms for the one-level densities of symmetric power L-functions in the level aspect, with Emmanuel Royer, published in Acta Arithmetica 141, no. 2, 153—170 (2010).
  
  In this paper, the lower order terms for the one-level densities of symmetric power L-functions in the level aspect are determined. The characters of the irreducible representations of SU(2) are the main ingredient.

• Comportement asymptotique des hauteurs des points de Heegner, with Nicolas Templier, published in Journal de Théorie des Nombres de Bordeaux 21, no. 3, 741—753 (2009).
  
  E is a fixed elliptic curve over the rational numbers. The purpose is to study the Néron-Tate height of Heegner points on E.We get an asymptotic formula for the Néron-Tate height of Heegner points on E on average over a subset of discriminants. The first and second order terms are obtained and a power saving in the remainder term is proved. Cancellations of Fourier coefficients of automorphic forms lie in the core of the proof.

• The second moment of Dirichlet twists of Hecke L-functions, with Peng Gao and Rizwanur Khan, published in Acta Arithmetica 140, 57—65 (2009).
  
  f is a fixed holomorphic cusp of level 1. The purpose is to study the asymptotic behaviour of the second moment of the twisted L-function of f by the primitive characters of conductor q as q goes to infinity. We get an asymptotic formula for the second moment of the twisted L-function of f by the primitive characters of conductor q with a polynomial saving in the error term and as q generically goes to infinity. This is a substancial improvement over Stefanicki’s previous result since his result holds for almost no q. It should be seen as an analogue of the fourth moment of Dirichlet L-functions.

• On the expected result for the second moment of twisted L-functions, published in Mathematisches Forschungsinstitut Oberwolfach, Report No. 33/2009, Explicit Methods in Number Theory (July 12th-July 18th), 14—16 (2009).
  
  Finding an asymptotic formula for the fourth moment of Dirichlet L-functions with a power saving in the error term should be philosophically as difficult as the corresponding question for the second moment of twisted L-functions. It turns out that the first problem has been completely solved by M. Young in the prime modulus case whereas the best result for the second one is only an asymptotic formula with a logarithmic saving in the error term so far. The purpose of this note is to identify the underlying analytic issue, which occurs in the second question.

• Hauteur asymptotique des points de Heegner, with Thomas Vidick, published in Canadian Journal of Mathematics, Vol. 60, No. 6, 1406—1436 (2008).
  
  E is a fixed elliptic curve over the rational numbers. The purpose is to study the Néron-Tate height of Heegner points on E. We get asymptotic formulas for the Néron-Tate height of Heegner points on E on average over a subset of discriminants: it is governed by the degree of the modular parametrisation of E as geometry suggests and the Néron-Tate height of traces of Heegner points on average over a subset of discriminants: we find a difference according to the rank of the elliptic curve. By Gross-Zagier formulas, it consists in proving asymptotic formulas for the first moments of the derivative of the Rankin-Selberg convolution of E with a certain weight one theta series attached to the principal ideal class of an imaginary quadratic field and the twisted L-function of E by a quadratic Dirichlet character. Many experimental results are discussed. These results give some insight to the problem of the discretisation of odd quadratic twists of elliptic curves.

• Statistics for low-lying zeros of Hecke L-functions in the level aspect, published in Mathematisches Forschungsinstitut Oberwolfach, Report No. 34/2007, Explicit Methods in Number Theory (July 15th-July 21st), 19--24 (2007).
  
  In this paper, we remind you of Iwaniec-Luo-Sarnak’s results on one-level densities for low-lying zeros of Hecke L-functions and Katz-Sarnak’s results on one-level densities for eigenvalues of orthogonal random matrices. Then, we introduce you to a very strange phenomenom discovered by Hughes and Miler: Mock-Gaussian behaviour.

• Real zeros and size of Rankin-Selberg L-functions in the level aspect, published in Duke Mathematical Journal, Vol. 131, No. 2 , 291—350 (2006).
  
  In this paper, some asymptotic formula is proved for the harmonic mollified second moment of a family of Rankin-Selberg L-functions. The main contribution is a substancial improvement of the admissible length of the mollifier which is done by solving a shifted convolution problem by a spectral method on average. Among the consequences are new subconvexity bounds, exponential decay of the analytic rank and non-vanishing results around the real axis for this family of L-functions.