zbMATH — the first resource for mathematics

Real zeros and size of Rankin-Selberg $$L$$-functions in the level aspect. (English) Zbl 1122.11059
This work is motivated by that of J. B. Conrey and K. Soundararajan [Invent. Math. 150, No. 1, 1–44 (2002; Zbl 1042.11053)], who showed that a positive proportion of quadratic Dirichlet $$L$$-functions have only trivial zeros on the real line. The author fixes a primitive form $$g$$ (satisfying certain conditions), and considers the $$L$$-series attached to the Rankin-Selberg convolution $$f\times g$$, where $$f$$ runs over a certain class of cusp forms, depending on $$g$$. It is then shown that the $$L$$-function has at most 8 nontrivial real zeros, for a positive proportion of such $$f$$. The proof depends on establishing asymptotic formulae for the mean-square (with respect to $$f)$$ of mollified versions of the $$L$$-function. The paper shows that one can use substantially longer mollifiers than was previously possible.

MSC:
 11M41 Other Dirichlet series and zeta functions 11F66 Langlands $$L$$-functions; one variable Dirichlet series and functional equations
Full Text:
References:
 [1] J. B. Conrey, D. W. Farmer, J. P. Keating, M. O. Rubinstein, and N. C. Snaith, Integral moments of $$L$$-functions , Proc. London Math. Soc. (3) 91 (2005), 33–104. · Zbl 1075.11058 [2] J. B. Conrey and K. Soundararajan, Real zeros of quadratic Dirichlet $$L$$-functions , Invent. Math. 150 (2002), 1–44. · Zbl 1042.11053 [3] J.-M. Deshouillers and H. Iwaniec, Kloosterman sums and Fourier coefficients of cusp forms , Invent. Math. 70 (1982/83), 219–288. · Zbl 0502.10021 [4] W. Duke, J. B. Friedlander, and H. Iwaniec, Bounds for automorphic $$L$$-functions, II , Invent. Math. 115 (1994), 219–239. · Zbl 0812.11032 [5] -, The subconvexity problem for Artin $$L$$-functions , Invent. Math. 149 (2002), 489–577. · Zbl 1056.11072 [6] A. Good, Cusp forms and eigenfunctions of the Laplacian , Math. Ann. 255 (1981), 523–548. · Zbl 0439.30031 [7] D. R. Heath-Brown and P. Michel, Exponential decay in the frequency of analytic ranks of automorphic $$L$$-functions , Duke Math. J. 102 (2000), 475–484. · Zbl 1166.11326 [8] J. Hoffstein and Lockhart, Coefficients of Maass forms and the Siegel zero , with an appendix “An effective zero-free region” by D. Goldfeld, J. Hoffstein, and D. Lieman, Ann. of Math. (2) 140 (1994), 161–181. JSTOR: · Zbl 0814.11032 [9] C. Hughes, Mollified and amplified moments of the Riemann zeta function: Some new theorems and conjectures , talk given at the workshop “Matrix Ensembles and $$L$$-functions,” Isaac Newton Institute for Mathematical Sciences, University of Cambridge, 2004, http://www.newton.cam.ac.uk/webseminars/pg$$+$$ws/2004/ rmaw05/0712/hughes/ [10] H. Iwaniec and E. Kowalski, Analytic Number Theory , Amer. Math. Soc. Colloq. Publ. 53 , Amer. Math. Soc., Providence, 2004. · Zbl 1059.11001 [11] H. Iwaniec, W. Luo, and P. Sarnak, Low lying zeros of families of $$L$$-functions , Inst. Hautes Études Sci. Publ. Math. 91 (2000), 55–131. · Zbl 1012.11041 [12] H. H. Kim, Functoriality for the exterior square of GL $$_4$$ and the symmetric fourth of GL$$_2$$, with an appendix 2 “Refined estimates towards the Ramanujan and Selberg conjectures” by H. H. Kim and P. Sarnak, J. Amer. Math. Soc. 16 (2003), 139–183. JSTOR: · Zbl 1018.11024 [13] H. H. Kim and F. Shahidi, Cuspidality of symmetric powers with applications , Duke Math. J. 112 (2002), 177–197. · Zbl 1074.11027 [14] E. Kowalski, The rank of the jacobian of modular curves: Analytic methods , Ph.D. dissertation, Rutgers University, New Brunswick, N.J., 1998, http://www.math.u-bordeaux1.fr/$$\sim$$kowalski/ [15] -, Dependency on the group in automorphic Sobolev inequalities , to appear in Forum Math., http://www.math.u-bordeaux1.fr/$$\sim$$kowalski/ [16] E. Kowalski and P. Michel, The analytic rank of $$J_0(q)$$ and zeros of automorphic $$L$$-functions , Duke Math. J. 100 (1999), 503–542. · Zbl 1161.11359 [17] E. Kowalski, P. Michel, and J. Vanderkam, Non-vanishing of high derivatives of automorphic $$L$$-functions at the center of the critical strip , J. Reine Angew. Math. 526 (2000), 1–34. · Zbl 1020.11033 [18] -, Rankin-Selberg $$L$$-functions in the level aspect , Duke Math. J. 114 (2002), 123–191. · Zbl 1035.11018 [19] B. KröTz and R. J. Stanton, Holomorphic extensions of representations, I: Automorphic functions , Ann. of Math. (2) 159 (2004), 641–724. JSTOR: · Zbl 1053.22009 [20] P. Michel, “Familles de fonctions $$L$$ de formes automorphes et applications” in Les XXIIèmes Journées Arithmétiques (Lille, Belgium, 2001) , J. Théor. Nombres Bordeaux 15 (2003), 275–307. · Zbl 1056.11027 [21] -, The subconvexity problem for Rankin-Selberg $$L$$-functions and equidistribution of Heegner points , Ann. of Math. (2) 160 (2004), 185–236. JSTOR: · Zbl 1068.11033 [22] G. Ricotta, Zéros réels et taille des fonctions $$L$$ de Rankin-Selberg par rapport au niveau , Ph.D. dissertation, Université Montpellier II, Montpellier, France, 2004, http://www.dms.umontreal.ca/$$\sim$$ricotta/research.htm P. Sarnak, Estimates for Rankin-Selberg $$L$$-functions and quantum unique ergodicity , J. Funct. Anal. 184 (2001), 419–453. · Zbl 1006.11022 [23] K. Soundararajan, The mollifier method and zeros of $$L$$-functions , talk given at the XXIIIrd Journées Arithmétiques, Graz, Austria, 2003. [24] G. N. Watson, A Treatise on the Theory of Bessel Functions , reprint of the 2nd (1944) ed., Cambridge Math. Lib., Cambridge Univ. Press, Cambridge, 1995. · Zbl 0063.08184 [25] A. Weil, On some exponential sums , Proc. Nat. Acad. Sci. U.S.A. 34 (1948), 204–207. · Zbl 0032.26102
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.