Bilinear forms with Kloosterman sums and applications

Kowalski, Emmanuel; Michel, Philippe; Sawin, Will

doi:10.4007/annals.2017.186.2.2

Cited by 51 publications

(92 citation statements)

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“…We recall that for individual values of a the limit of our current methods is q ď x 2{3´ε for an arbitrary fixed ε ą 0 . Our method builds on an approach due to Blomer (2008) based on the Voronoi summation formula which we combine with some recent results on bilinear sums of Kloosterman sums due Kowalski, Michel and Sawin (2017) and Shparlinski (2017). We also make use of extra applications of the Voronoi summation formulae after expanding into Kloosterman sums and this reduces the problem to estimating the number of solutions to multiplicative congruences.…”

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confidence: 99%

Bilinear sums of Kloosterman sums, multiplicative congruences and average values of the divisor function over families of arithmetic progressions

Kerr

Shparlinski

2020

Res. number theory

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We obtain several asymptotic formulas for the sum of the divisor function τ pnq with n ď x in an arithmetic progressions n " a pmod qq on average over a from a set of several consecutive elements from set of reduced residues modulo q and on average over arbitrary sets. The main goal is to obtain nontrivial result for q ě x 2{3 with the small amount of averaging over a. We recall that for individual values of a the limit of our current methods is q ď x 2{3´ε for an arbitrary fixed ε ą 0 . Our method builds on an approach due to Blomer (2008) based on the Voronoi summation formula which we combine with some recent results on bilinear sums of Kloosterman sums due Kowalski, Michel and Sawin (2017) and Shparlinski (2017). We also make use of extra applications of the Voronoi summation formulae after expanding into Kloosterman sums and this reduces the problem to estimating the number of solutions to multiplicative congruences. nďX n"a pmod qq τ pnq.

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Bilinear sums of Kloosterman sums, multiplicative congruences and average values of the divisor function over families of arithmetic progressions

Kerr

Shparlinski

2020

Res. number theory

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“…This technique goes back to Karatsuba and Vinogradov (for the function x → χ(x + 1)). It has been also used by Friedlander-Iwaniec [FI85] (for the function x → e x q ), Fouvry-Michel [FM98] and Kowalski-Michel-Sawin [KMS17,KMS18].…”

Section: Advanced Completions Methods: the +Ab Shiftmentioning

confidence: 99%

“…where (α m ) m∼M , (β n ) n∼N are sequences of complex numbers of modulus bounded by 1. We leave it to the interested reader to fill in the details (or to look at [FM98,KMS17] or [KMS18]). The first step is to apply Cauchy-Schwarz to smooth out the n variable: for a suitable smooth function V , compactly supported in [1/2, 5/2] and bounded by 1, one has m∼M,n∼N α m β n K(mn) N 1/2 m1,m2∼M α m1 α m2 n V ( n N )K(m 1 n)K(m 2 n) 1/2 .…”

Section: Advanced Completions Methods: the +Ab Shiftmentioning

confidence: 99%

“…(a, m 1 , m 2 , n) ∈ [A, 2A[×[M, 2M [ 2 ×[N, 2N [←→ (an, am 1 , am 2 ) (mod q) = (r, s 1 , s 2 ) ∈ F 3 q . Considering the fiber counting function for that map, namely ν(r, s 1 , s 2 ) := (a,n,m1,m2) a∼A,|n| 2N, mi≃M α m1 α m2 δ an=r, ami=si (mod q)one shows that for AN < q/2 one has (r,s1,s2)∈Fq 3 |ν(r, s 1 , s 2 )| ≪ AM 2 N,(r,s1,s2)∈Fq 3 |ν(r, s 1 , s 2 )| 2 q o(1) AM 2 N.Applying Hölder's inequality leads us to the problem of bounding the following complete sum indexed by the parameter b(16.7) r∈Fq |R(r, b)| 2 − q r∈Fq |K(r, b)| 2 .16 If N 1 2 q 1/2+1/2l the Pólya-Vinogradov inequality is non trivial already.-In the case of the Kloosterman sums K(n) = Kl 2 (n; q), the bound established in[KMS17] together with [BM15,BFK + 17] leads to an asymptotic formula for the second moment of character twists of modular L-functions: for f a fixed primitive cusp form, one has 1 q − 1χ (mod q) |L(f ⊗ χ, 1/2)| 2 = M T f (log q) + O f (q −1/145 ) for q prime, where M T f (log q) isa polynomial in log q (of degree 1) depending on f . This completes the work of Young for f an Eisenstein series [You11] and of Blomer-Milicevic for f cuspidal and q suitably composite [BM15].…”

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Lectures on applied ℓ-adic cohomology

Fouvry¹,

Kowalski²,

Michel³

et al. 2019

Contemporary Mathematics

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We describe how a systematic use of the deep methods from ℓ-adic cohomology pioneered by Grothendieck and Deligne and further developed by Katz and Laumon help make progress on various classical questions from analytic number theory. This text is an extended version of a series of lectures given during the 2016 Arizona Winter School.2000 Mathematics Subject Classification. Primary . 1In number theory, especially analytic number theory, one is interested in studying the behaviour of some given arithmetic function along congruence classes, for instance to determine whether a set of integers has finite or infinite intersection with some congruence class. The analysis of such problem, which may involve quite sophisticated manipulations, often involves certain specific classes of functions on Z/qZ.When studying such functions, it is natural to invoke the Chinese Remainder Theoremwhich largely reduces the study to the case of prime power moduli; then, in many instances, the deepest case is when q is a prime; the ring Z/qZ is then a finite field, denoted F q , and often the functions that occur are what we will call trace functions.The objective of these lectures is utilitarian: our aim is to describe these trace functions, many examples, their theory and most importantly how they are handled when they occur in analytic number theory. Indeed the mention of "étale" or "ℓ-adic cohomology", "sheaves", "purity", "functors", "local systems" or "vanishing cycles" sounds forbidding to the working analytic number theorist and often prevents him/her to embrace the subject and make full use of the powerful methods that Deligne, Katz, Laumon have developed for us. It is our hope that after these introductory lectures, any of the remaining readers will feel ready for and at ease with more serious activities such as the reading of the wonderful series of orange books by Katz, and eventually will be able to tackle by him/herself any trace function that nature has laid in front of him/her.Acknowledgements. These expository notes are an expanded version of a series of lectures given by Ph.M. and W.S. during the 2016 Arizona Winter School and based on our recent joint works.We would like to thank the audience for its attention and its numerous questions during the daily lectures, as well as the teams of student, who engaged in the research activities that we proposed during the evening sessions, for their enthusiasm. Big thanks are also due to Alina Bucur, Bryden Cais and David Zureick-Brown for the perfect organization, making this edition of the AWS a memorable experience. We would also like to thank the referees for correcting many mistakes and typosin earlier versions of this text.

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“…So we have chosen to avoid the completion step, in order to simplify an already complex argument. It should be noted however that, for small values of l, the completion step is worth pursuing, and that is was crucial in [KMS17] to obtain non-trivial bounds for l " 2 (which was the only case that could be handled in [KMS17], because, as noted earlier, the diagonal variety in that paper was of codimension 1).…”

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Stratification and averaging for exponential sums: bilinear forms with generalized Kloosterman sums

Kowalski¹,

Michel²,

Sawin³

2020

Annali Scuola Normale Superiore - Classe Di Scienze

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We introduce a new comparison principle for exponential sums over finite fields in order to study "sum-product" sheaves that arise in the study of general bilinear forms with coefficients given by trace functions modulo a prime q. When these functions are hyper-Kloosterman sums with characters, we succeed in establishing cases of this principle that lead to non-trivial bounds below the Pólya-Vinogradov range. This property is proved by a subtle interplay betweenétale cohomology in its algebraic and diophantine incarnations. We give a first application of our bilinear estimates concerning the first moment of a family of L-functions of degree 3.When facing such problems, one is often led to the problem of bounding non-trivially some bilinear forms BpK, α, βq " ÿ ÿ mďM,nďN α m β n Kpmn; qq,where the ranges of the variables M, N ě 1 usually depend on q, and α " pα m q mďM , β " pβ n q nďN are complex numbers which, depending on the initial problem, are quite arbitrary. One of the main objectives is to improve on the trivial boundfor ranges of M and N that are as small as possible compared to q; indeed, this uniformity is often more important than the strength of the saving compared to the trivial bound. A natural benchmark is the Pólya-Vinogradov method, which often provides non-trivial bounds as long as M, N ě q 1{2 . Indeed, obtaining a result below that range is usually extremely challenging. When the modulus q is composite, a number of techniques exploiting the possibility of factoring q (starting with the Chinese Remainder Theorem) become available, and results exist in fair generality.In this paper, we will only consider the case where q is a prime, and when K is a trace function (see [FKM14b] for a background survey).The landmark result in this setting is the work of Burgess [Bur62], which provides a non-trivial bound for the sum ÿ nďN χpnq when χ is a non-trivial Dirichlet character modulo q and N ě q 3{8`η , for any η ą 0. This is therefore well below Pólya-Vinogradov range. The ideas of Burgess (especially the "`ab shifting trick") combine successfully the multiplicativity of χ and the (almost) invariance of intervals by additive translations. Another twist of Burgess's method was given by the works of Karatsuba and Vinogradov, Friedlander-Iwaniec [FI85] and subsequently Fouvry-Michel [FM98] to bound non-trivially the bilinear sums BpK, α, βq for various choices of functions K and ranges M, N shorter than q 1{2 . In particular, using some version of the Sato-Tate equidistribution laws due to Katz [Kat90], Fouvry and Michel considered Kpx; qq " e´x k`a x q¯, k P Z´t0, 1, 2u, a P Fq , px, qq " 1, and proved that for any δ ą 0, there exists η ą 0 such that,as long asThe condition M N ě q 3{4`δ is believed to be a barrier in this setting analogous to the condition N ą q 1{4`δ in the Burgess bound for short character sums.In our previous paper [KMS17], motivated by the study of moments of L-functions (especially in our papers with Blomer, Milićević and Fouvry [BFK`17, BFK`]), we obtained bounds of type (1.1) whe...

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Bilinear forms with Kloosterman sums and applications

Cited by 51 publications

References 39 publications

Bilinear sums of Kloosterman sums, multiplicative congruences and average values of the divisor function over families of arithmetic progressions

Bilinear sums of Kloosterman sums, multiplicative congruences and average values of the divisor function over families of arithmetic progressions

Lectures on applied ℓ-adic cohomology

Stratification and averaging for exponential sums: bilinear forms with generalized Kloosterman sums

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