Subconvexity for twisted L-functions over number fields via shifted convolution sums

详细信息    查看全文

• 作者：P. Maga
• 关键词：Mathematics Subject Classificationprimary 11F41 ; 11M41 ; secondary 11F72
• 刊名：Acta Mathematica Hungarica
• 出版年：2017
• 出版时间：February 2017
• 年：2017
• 卷：151
• 期：1
• 页码：232-257
• 全文大小：
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics, general;
• 出版者：Springer Netherlands
• ISSN：1588-2632
• 卷排序：151

文摘

Assume that \({\pi}\) is a cuspidal automorphic \({{\rm GL}_{2}}\) representation over a number field F. Then for any Hecke character \({\chi}\) of conductor \({\mathfrak{q}}\), the subconvex bound$$L(1/2,\pi \otimes \chi) \ll_{F,\pi,\chi_{\infty},\varepsilon} \mathcal{N}{\mathfrak{q}}^{3/8+\theta/4+\varepsilon}$$holds for any \({\varepsilon > 0}\), where \({\theta}\) is any constant towards the Ramanujan-Petersson conjecture (\({\theta = 7/64}\) is admissible). In these notes, we derive this bound from the spectral decomposition of shifted convolution sums worked out by the author in [21].Mathematics Subject Classificationprimary 11F4111M41secondary 11F72Key words and phrasessubconvexityshifted convolution sumReferences1.Blomer V., Brumley F.: On the Ramanujan conjecture over number fields. Ann. Math., 174, 581–605 (2011)MathSciNetCrossRefMATHGoogle Scholar2.Blomer V., Harcos G.: The spectral decomposition of shifted convolution sums. Duke Math. J. 144, 321–339 (2008)MathSciNetCrossRefMATHGoogle Scholar3.V. Blomer and G. Harcos, Twisted L-functions over number fields and Hilbert’s eleventh problem, Geom. Funct. Anal., 20 (2010), 1–52; Erratum: www.renyi.hu/~gharcos/hilbert_erratum.pdf.4.V. Blomer, G. Harcos and P. Michel, A Burgess-like subconvex bound for twisted L-functions, Forum Math., 19 (2007), 61–105, Appendix 2 by Z. Mao.5.Bruggeman R. V., Miatello R. J.: Sum formula for SL₂ over a number field and Selberg type estimate for exceptional eigenvalues. Geom. Funct. Anal. 8, 627–655 (1998)MathSciNetCrossRefMATHGoogle Scholar6.Bruggeman R. V., Miatello R. J., Pacharoni I.: Estimates for Kloosterman sums for totally real number fields. J. Reine Angew. Math. 535, 103–164 (2001)MathSciNetMATHGoogle Scholar7.Bruggeman R. V., Motohashi Y.: Sum formula for Kloosterman sums and fourth moment of the Dedekind zeta-function over the Gaussian number field. Funct. Approx. Comment. Math. 31, 23–92 (2003)MathSciNetMATHGoogle Scholar8.D. Bump, Automorphic Forms and Representations, Cambridge Studies in Advanced Mathematics 55, Cambridge University Press, (Cambridge, 1997).9.Burgess D. A.: On character sums and L-series. II. Proc. London Math. Soc. (3) 13, 524–536 (1963)MathSciNetCrossRefMATHGoogle Scholar10.V. A. Bykovskiĭ, A trace formula for the scalar product of Hecke series and its applications, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI), 226 (1996), (Anal. Teor. Chisel i Teor. Funktsii. 13), 14–36, 235–236; translation in J. Math. Sci. (New York), 89 (1998), 915–932.11.Conrey J. B., Iwaniec H.: The cubic moment of central values of automorphic L-functions. Ann. Math. 151, 1175–1216 (2000)MathSciNetCrossRefMATHGoogle Scholar12.Cohen P. B.: Hyperbolic equidistribution problems on Siegel 3-folds and Hilbert modular varieties. Duke Math. J. 129, 87–127 (2005)MathSciNetCrossRefMATHGoogle Scholar13.J. W. Cogdell, I. Piatetski-Shapiro and P. Sarnak, Estimates on the critical line for Hilbert modular L-functions and applications, preprint, 2001.14.Duke W.: Hyperbolic distribution problems and half-integral weight Maass forms. Invent. Math. 92, 73–90 (1988)MathSciNetCrossRefMATHGoogle Scholar15.W. Duke, J. Friedlander and H. Iwaniec, Bounds for automorphic L-functions, Invent. Math., 112 (1993), 1–8.16.Duke W., Schulze-Pillot R.: Representation of integers by positive ternary quadratic forms and equidistribution of lattice points on ellipsoids. Invent. Math. 99, 49–57 (1990)MathSciNetCrossRefMATHGoogle Scholar17.I. S. Gradshteyn and I. M. Ryzhik, Table of integrals, series, and products, Seventh edition, Elsevier Academic Press (Amsterdam, 2007); seventh edition, translated from the Russian, translation edited and with a preface by Daniel Zwillinger and Victor Moll.18.Harcos G.: Uniform approximate functional equation for principal L-functions. Int. Math. Res. Not. 18, 923–932 (2002)MathSciNetCrossRefMATHGoogle Scholar19.Iwaniec H.: Fourier coefficients of modular forms of half-integral weight Invent. Math. 87, 385–401 (1987)MathSciNetMATHGoogle Scholar20.H. Iwaniec and P. Sarnak, Perspectives on the analytic theory of L-functions, GAFA 2000 (Tel Aviv, 1999), Geom. Funct. Anal. (2000), Special Volume, Part II, 705–741.21.P. Maga, The spectral decomposition of shifted convolution sums over number fields, J. Reine Angew. Math. (to appear).22.Maga P.: A semi-adelic Kuznetsov formula over number fields. Int. J. Number Theory 9, 1649–1681 (2013)MathSciNetCrossRefMATHGoogle Scholar23.P. Maga, Subconvexity and shifted convolution sums over number fields, Central European University (Budapest, 2013).24.Michel P., Venkatesh A.: The subconvexity problem for GL₂. Publ. Math. Inst. Hautes Études Sci. 111, 171–271 (2010)CrossRefGoogle Scholar25.Molteni G.: Upper and lower bounds at \({s = 1}\) for certain Dirichlet series with Euler product. Duke Math. J. 111, 133–158 (2002)MathSciNetCrossRefMATHGoogle Scholar26.W. Narkiewicz, Elementary and Analytic Theory of Algebraic Numbers, Monografie Matematyczne, Tom. 57, PWN—Polish Scientific Publishers, (Warsaw, 1974).27.J. Neukirch, Algebraic Number Theory, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 322, Springer-Verlag (Berlin, 1999); translated from the 1992 German original and with a note by N. Schappacher, with a foreword by G. Harder.28.Venkatesh A.:“Beyond endoscopy” and special forms on GL(2). J. Reine Angew. Math. 577, 23–80 (2004)MathSciNetMATHGoogle Scholar29.Venkatesh A.: Sparse equidistribution problems, period bounds and subconvexity. Ann. Math. 172, 989–1094 (2010)MathSciNetCrossRefMATHGoogle Scholar30.Wu H.: Burgess-like subconvex bounds for \({{\rm GL}_{2} \times {\rm GL}_{1}}\). Geom. Funct. Anal. 24, 968–1036 (2014)MathSciNetCrossRefGoogle Scholar31.Zhang S.-W.: Equidistribution of CM-points on quaternion Shimura varieties. Int. Math. Res. Not. 59, 3657–3689 (2005)MathSciNetCrossRefMATHGoogle ScholarCopyright information© Akadémiai Kiadó, Budapest, Hungary 2016Authors and AffiliationsP. Maga1Email author1.Alfréd Rényi Institute of Mathematics, Hungarian Academy of SciencesBudapestHungary About this article CrossMark Publisher Name Springer Netherlands Print ISSN 0236-5294 Online ISSN 1588-2632 About this journal Reprints and Permissions Article actions .buybox { margin: 16px 0 0; position: relative; } .buybox { font-family: Source Sans Pro, Helvetica, Arial, sans-serif; font-size: 14px; font-size: .875rem; } .buybox { zoom: 1; } .buybox:after, .buybox:before { content: ''; display: table; } .buybox:after { clear: both; } /*---------------------------------*/ .buybox .buybox__header { border: 1px solid #b3b3b3; border-bottom: 0; padding: 8px 12px; position: relative; background-color: #f2f2f2; } .buybox__header .buybox__login { font-family: Source Sans Pro, Helvetica, Arial, sans-serif; font-size: 14px; font-size: .875rem; letter-spacing: .017em; display: inline-block; line-height: 1.2; padding: 0; } .buybox__header .buybox__login:before { position: absolute; top: 50%; -webkit-transform: perspective(1px) translateY(-50%); transform: perspective(1px) translateY(-50%); content: '\A'; width: 34px; height: 34px; left: 10px; } /*---------------------------------*/ .buybox .buybox__body { padding: 0; padding-bottom: 16px; position: relative; text-align: center; background-color: #fcfcfc; border: 1px solid #b3b3b3; } .buybox__body .buybox__section { padding: 16px 12px 0 12px; text-align: left; } .buybox__section .buybox__buttons { text-align: center; width: 100%; } /********** mycopy buybox specific **********/ .buybox.mycopy__buybox .buybox__section .buybox__buttons { border-top: 0; padding-top: 0; } /******/ .buybox__section:nth-child(2) .buybox__buttons { border-top: 1px solid #b3b3b3; padding-top: 20px; } .buybox__buttons .buybox__buy-button { display: inline-block; text-align: center; margin-bottom: 5px; padding: 6px 12px; } .buybox__buttons .buybox__price { white-space: nowrap; text-align: center; font-size: larger; padding-top: 6px; } .buybox__section .buybox__meta { letter-spacing: 0; padding-top: 12px; } .buybox__section .buybox__meta:only-of-type { padding-top: 0; position: relative; bottom: 6px; } /********** mycopy buybox specific **********/ .buybox.mycopy__buybox .buybox__section .buybox__meta { margin-top: 0; margin-bottom: 0; } /******/ .buybox__meta .buybox__product-title { display: inline; font-weight: bold; } .buybox__meta .buybox__list { line-height: 1.3; } .buybox__meta .buybox__list li { position: relative; padding-left: 1em; list-style: none; margin-bottom: 5px; } .buybox__meta .buybox__list li:before { font-size: 1em; content: '\2022'; float: left; position: relative; top: .1em; font-family: serif; font-weight: 600; text-align: center; line-height: inherit; color: #666; width: auto; margin-left: -1em; } .buybox__meta .buybox__list li:last-child { margin-bottom: 0; } /*---------------------------------*/ .buybox .buybox__footer { border: 1px solid #b3b3b3; border-top: 0; padding: 8px 12px; position: relative; border-style: dashed; } /*-----------------------------------------------------------------*/ @media screen and (min-width: 460px) and (max-width: 1074px) { .buybox__body .buybox__section { display: inline-block; vertical-align: top; padding: 12px 12px; padding-bottom: 0; text-align: left; width: 48%; } .buybox__body .buybox__section { padding-top: 16px; padding-left: 0; } .buybox__section:nth-of-type(2) .buybox__meta { border-left: 1px solid #d3d3d3; padding-left: 28px; } .buybox__section:nth-of-type(2) .buybox__buttons { border-top: 0; padding-top: 0; padding-left: 16px ; } .buybox__buttons .buybox__buy-button { } /********** article buybox specific **********/ .buybox.article__buybox .buybox__section:nth-of-type(2) { margin-top: 16px; padding-top: 0; } .buybox.article__buybox .buybox__section:nth-of-type(2) .buybox__meta { margin-top: 40px; padding-top: 0; padding-bottom: 45px; } .buybox.article__buybox .buybox__section:nth-of-type(2) .buybox__meta:only-of-type { margin-top: 8px; padding-top: 12px; padding-bottom: 12px; } /********** mycopy buybox specific **********/ .buybox.mycopy__buybox .buybox__section:first-child { width: 69%; } .buybox.mycopy__buybox .buybox__section:last-child { width: 29%; } /******/ } /*-----------------------------------------------------------------*/ @media screen and (max-width: 459px) { /********** mycopy buybox specific **********/ .buybox.mycopy__buybox .buybox__body { padding-bottom: 5px; } .buybox.mycopy__buybox .buybox__section:last-child { text-align: center; width: 100%; } .buybox.mycopy__buybox .buybox__buttons { display: inline-block; width: 150px ; } /******/ } /*-----------------------------------------------------------------*/ Log in to check access Buy (PDF) EUR 34,95 Unlimited access to the full article Instant download Include local sales tax if applicable Find out about institutional subscriptions (function () { var forEach = function (array, callback, scope) { for (var i = 0; i Export citation .RIS Papers Reference Manager RefWorks Zotero .ENW EndNote .BIB BibTeX JabRef Mendeley Share article Email Facebook Twitter LinkedIn Cookies We use cookies to improve your experience with our site. More information Accept Over 10 million scientific documents at your fingertips