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Subconvexity bounds for automorphic L-functions

Part of: Discontinuous groups and automorphic forms Algebraic number theory: global fields Zeta and $L$-functions: analytic theory

Published online by Cambridge University Press:  23 July 2009

A. Diaconu  and
P. Garrett
A. Diaconu
Affiliation:
School of Mathematics, University of Minnesota, Minneapolis, MN 55455, USA ([email protected])
P. Garrett
Affiliation:
School of Mathematics, University of Minnesota, Minneapolis, MN 55455, USA ([email protected])
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Abstract

We break the convexity bound in the t-aspect for L-functions attached to cusp forms f for GL2(k) over arbitrary number fields k. The argument uses asymptotics with error term with a power saving, for second integral moments over spectral families of twists L(s,f⊗χ) by Grossencharacters χ, from our previous paper on integral moments.

Keywords

subconvexityL-functionsintegral momentsPoincaré seriesEisenstein seriesspectral decompositionmeromorphic continuation

MSC classification

Secondary: 11R42: Zeta functions and $L$-functions of number fields 11F66: Langlands $L$-functions; one variable Dirichlet series and functional equations 11F67: Special values of automorphic $L$-series, periods of modular forms, cohomology, modular symbols 11F70: Representation-theoretic methods; automorphic representations over local and global fields 11M41: Other Dirichlet series and zeta functions 11R47: Other analytic theory
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 9 , Issue 1 , January 2010 , pp. 95 - 124
Copyright
Copyright © Cambridge University Press 2010

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References

1.Bernstein, J. and Reznikov, A., Analytic continuation of representations and estimates of automorphic forms, Annals Math. 150 (1999), 329–352.CrossRefGoogle Scholar
2.Bernstein, J. and Reznikov, A., Periods, subconvexity of L-functions, and representation theory, J. Diff. Geom. 10 (2005), 129–141.Google Scholar
3.Blomer, V., Harcos, G. and Michel, P., A Burgess-like subconvexity bound for twisted L-functions, Forum Math. 19 (2007), 61–105.CrossRefGoogle Scholar
4.Burgess, D., On character sums and L-series, II, Proc. Lond. Math. Soc. 313 (1963), 24–36.Google Scholar
5.Casselman, W., On some results of Atkin and Lehner, Math. Annalen 201 (1973), 301–314.CrossRefGoogle Scholar
6.Chandrasekharan, K., Introduction to analytic number theory, Die Grundlehren der mathematischen Wissenschaften, Band 148 (Springer, 1968).CrossRefGoogle Scholar
7.Cogdell, J., Piatetski-Shapiro, I. and Sarnak, P., Estimates for Hilbert modular L-functions and applications, in preparation.Google Scholar
8.Conrey, J. B. and Iwaniec, H., The cubic moment of central values of automorphic L-functions, Annals Math. (2) 151 (2000), 1175–1216.CrossRefGoogle Scholar
9.Davenport, H., Multiplicative number theory (Springer, 2000).Google Scholar
10.Diaconu, A. and Garrett, P., Integral moments of automorphic L-functions, J. Inst. Math. Jussieu 8(2) (2009), 335–382.CrossRefGoogle Scholar
11.Diaconu, A. and Goldfeld, D., Second moments of GL2 automorphic L-functions, in Analytic Number Theory, Proc. of the Gauss–Dirichlet Conf., Göttingen, 2005, Clay Mathematical Proceedings, pp. 77–105 (American Mathematical Society, Providence, RI, 2005).Google Scholar
12.Diaconu, A. and Goldfeld, D., Second moments of quadratic Hecke L-series and multiple Dirichlet series, I, in Multiple Dirichlet series, automorphic forms, and analytic number theory, Proceedings of Symposia in Pure Mathematics, Volume 75, pp. 59–89 (American Mathematical Society, Providence, RI, 2006).CrossRefGoogle Scholar
13.Diaconu, A., Garrett, P. and Goldfeld, D., Moments for L-functions for GLr × GL r−1, in preparation.Google Scholar
14.Donnelly, H., On the cuspidal spectrum for finite volume symmetric spaces, J. Diff. Geom. 17 (1982), 239–253.Google Scholar
15.Duke, W., Friedlander, J. and Iwaniec, H., Bounds for automorphic L-functions, Invent. Math. 112 (1993), 1–18.CrossRefGoogle Scholar
16.Duke, W., Friedlander, J. and Iwaniec, H., A quadratic divisor problem, Invent. Math. 115 (1994), 209–217.CrossRefGoogle Scholar
17.Duke, W., Friedlander, J. and Iwaniec, H., Bounds for automorphic L-functions, II, Invent. Math. 115 (1994), 219–239.CrossRefGoogle Scholar
18.Duke, W., Friedlander, J. and Iwaniec, H., The subconvexity problem for Artin L-functions, Invent. Math. 149 (2000), 489–577.CrossRefGoogle Scholar
19.Duke, W., Friedlander, J. and Iwaniec, H., Bounds for automorphic L-functions, III, Invent. Math. 143 (2001), 221–248.CrossRefGoogle Scholar
20.Good, A., The square mean of Dirichlet series associated with cusp forms, Mathematika 29 (1982), 278–295.CrossRefGoogle Scholar
21.Good, A., The convolution method for Dirichlet series, in The Selberg Trace Formula and Related Topics, Brunswick, Maine, 1984, Contemporary Mathematics, Volume 53, pp. 207–214 (American Mathematical Society, Providence, RI, 1986).CrossRefGoogle Scholar
22.Gradshteyn, I. S. and Ryzhik, I. M., Table of integrals, series, and products, 5th edn (Academic Press, 1994).Google Scholar
23.Harcos, G. and Michel, P., The subconvexity problem for Rankin–Selberg L-functions and equidistribution of Heegner points, II, Invent. Math. 163 (2006), 581–655.CrossRefGoogle Scholar
24.Heath-Brown, D. R., Hybrid bounds for Dirichlet L-functions, Invent. Math. 47 (1978), 149–170.CrossRefGoogle Scholar
25.Heath-Brown, D. R., The growth rate of the Dedekind zeta-function on the critical line, Acta Arith. 49 (1988), 323–339.CrossRefGoogle Scholar
26.Hoffstein, J. and Lockhart, P., Coefficients of Maass forms and the Siegel zero, Annals Math. 140 (1994), 161–181.CrossRefGoogle Scholar
27.Hoffstein, J. and Ramakrishnan, D., Siegel zeros and cusp forms, Int. Math. Res. Not. 6 (1995), 279–308.CrossRefGoogle Scholar
28.Ivic, A. and Motohashi, Y., The mean square of the error term for the fourth power moment of the zeta-function, Proc. Lond. Math. Soc. 69 (1994), 309–329.CrossRefGoogle Scholar
29.Iwaniec, H. and Sarnak, P., Perspectives on the analytic theory of L-functions, Geom. Funct. Analysis Special Volume (2000), 705–741.Google Scholar
30.Kaufman, R. M., Estimate of the Hecke L-functions on the half line, J. Soviet Math. 17(5) (1981), 2107–2115.CrossRefGoogle Scholar
31.Kim, H., On local L-functions and normalized intertwining operators, Can. J. Math. 57 (2005), 535–597.CrossRefGoogle Scholar
32.Kim, H. and Shahidi, F., Cuspidality of symmetric powers with applications, Duke Math. J. 112 (2002), 177–197.CrossRefGoogle Scholar
33.Kowalski, E., Michel, P. and Vanderkam, J., Rankin–Selberg L-functions in the level aspect, Duke Math. J. 114 (2002), 123–191.CrossRefGoogle Scholar
34.Krötz, B. and Stanton, R., Holomorphic extensions of representations, I, Automorphic functions, Annals Math. (2) 159 (2004), 641–724.CrossRefGoogle Scholar
35.Lau, Y.-K., Liu, J. and Ye, Y., Subconvexity bounds for Rankin–Selberg L-functions for congruence subgroups, J. Number Theory 121 (2006), 204–223.CrossRefGoogle Scholar
36.Letang, D., Subconvex bounds in conductor-depth aspect for automorphic L-functions on GL(2), preprint arXiv:0904.1028 (2009).Google Scholar
37.Li, X., Bounds for GL 3 × GL 2L-functions and GL 3L-functions, Annals Math., in press.Google Scholar
38.Lindenstrauss, E. and Venkatesh, A., Existence and Weyl's law for spherical cusp forms, Geom. Funct. Analysis 17 (2007), 220–251.CrossRefGoogle Scholar
39.Meurman, T., On the order of Maass L-functions on the critical line, in Number Theory, Volume I, Budapest Colloquium, 1987, Volume 51, pp. 325–354 (Mathematical Society Bolyai, Budapest, 1990).Google Scholar
40.Michel, P., The subconvexity problem for Rankin–Selberg L-functions and equidistribution of Heegner points, Annals Math. (2) 160 (2004), 185–236.CrossRefGoogle Scholar
41.Michel, P., Analytic number theory and families of automorphic L-functions, in Automorphic forms and applications, IAS/Park City Mathematics Series, Volume 12, pp. 181–295 (American Mathematical Society, Providence, RI, 2007).CrossRefGoogle Scholar
42.Michel, P. and Venkatesh, A., Equidistribution, L-functions, and ergodic theory: on some problems of Yu. Linnik, International Congress of Mathematicians, Volume II, pp. 421–457 (European Mathematical Society, Zurich, 2006).Google Scholar
43.Petridis, Y. and Sarnak, P., Quantum unique ergodicity for SL 2()\H 3 and estimates for L-functions, J. Evol. Eqns 1 (2001), 277–290.CrossRefGoogle Scholar
44.Sarnak, P., Fourth moments of Grössencharakteren zeta functions, Commun. Pure Appl. Math. 38 (1985), 167–178.CrossRefGoogle Scholar
45.Sarnak, P., Integrals of products of eigenfunctions, Int. Math. Res. Not. 1994 (1994), 251–260.CrossRefGoogle Scholar
46.Sarnak, P., Arithmetic quantum chaos, in The Schur Lectures, Tel Aviv, 1992, Israel Math. Conf., Bar Ilan, Volume 8, pp. 183–236 (1995).Google Scholar
47.Sarnak, P., Estimates for Rankin–Selberg L-functions and quantum unique ergodicity, J. Funct. Analysis 184 (2001), 419–453.CrossRefGoogle Scholar
48.Selberg, A., On the estimation of Fourier coefficients of modular forms, in Number theory, Proceedings of Symposia in Pure Mathematics, Volume 8, pp. 1–15 (American Mathematical Society, Providence, RI, 1965)Google Scholar
49.Söhne, P., An upper bound for the Hecke zeta-functions with Grossencharacters, J. Number Theory 66 (1997), 225–250.CrossRefGoogle Scholar
50.Truelsen, J. L., Quantum unique ergodicity for Eisenstein series on the Hilbert modular group over a totally real field, preprint arXiv:math/0706.4239v3 (2007; revised 2008).Google Scholar
51.Venkatesh, A., Sparse equidistribution problems, period bounds, and subconvexity, Annals Math., in press.Google Scholar
52.Weyl, H., Zur Abschatzung von ζ(1 + it), Math. Z. 10 (1921), 88–101.CrossRefGoogle Scholar
53.Zhang, Q., Integral mean values of Maass L-functions, Int. Math. Res. Not. 2006 (2006), article 41417.CrossRefGoogle Scholar