Duke Mathematical Journal

Rankin-Selberg L-functions in the level aspect

E. Kowalski, P. Michel, and J. VanderKam

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Abstract

In this paper we calculate the asymptotics of various moments of the central values of Rankin-Selberg convolution L-functions of large level, thus generalizing the results and methods of W. Duke, J. Friedlander, and H. Iwaniec and of the authors. Consequences include convexity-breaking bounds, nonvanishing of a positive proportion of central values, and linear independence results for certain Hecke operators.

Article information

Source
Duke Math. J. Volume 114, Number 1 (2002), 123-191.

Dates
First available in Project Euclid: 18 June 2004

Permanent link to this document
http://projecteuclid.org/euclid.dmj/1087575359

Mathematical Reviews number (MathSciNet)
MR1 915 038

Digital Object Identifier
doi:10.1215/S0012-7094-02-11416-1

Zentralblatt MATH identifier
1035.11018

Subjects
Primary: 11F66: Langlands $L$-functions; one variable Dirichlet series and functional equations
Secondary: 11G40: $L$-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture [See also 14G10]

Citation

Kowalski, E.; Michel, P.; VanderKam, J. Rankin-Selberg L -functions in the level aspect. Duke Mathematical Journal 114 (2002), no. 1, 123--191. doi:10.1215/S0012-7094-02-11416-1. http://projecteuclid.org/euclid.dmj/1087575359.


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References

  • A. Abbes and E. Ullmo, Comparaison des métriques d'Arakelov et de Poincaré sur $X_0(N)$, Duke Math J. 80 (1995), 295--307.
  • A. O. L. Atkin and J. Lehner, Hecke operators on $\Gamma_0(m)$, Math. Ann. 185 (1970), 134--160.
  • A. O. L. Atkin and W. C. W. Li, Twists of newforms and pseudo-eigenvalues of $W$-operators, Invent. Math. 48 (1978), 221--243.
  • M. Bertolini and H. Darmon, A rigid analytic Gross-Zagier formula and arithmetic applications, Ann. of Math. (2) 146 (1997), 111--147.
  • A. Brumer, ``The rank of $J\sb 0(N)$'' in Columbia University Number Theory Seminar (New York, 1992), Asterisque 228, Soc. Math. France, Montrouge, 1995, 41--68.
  • D. Bump, Automorphic Forms and Representations, Cambridge Stud. Adv. Math. 55, Cambridge Univ. Press, Cambridge, 1997.
  • D. Bump, S. Friedberg, and J. Hoffstein, Nonvanishing theorems for $L$-functions of modular forms and their derivatives, Invent. Math. 102 (1990), 543--618.
  • J. B. Conrey and H. Iwaniec, The cubic moment of central values of automorphic $L$-functions, Ann. of Math. (2) 151 (2000), 1175--1216.
  • H. Davenport, Multiplicative Number Theory, Grad. Texts in Math. 74, Springer, New York, 1980.
  • J.-M. Deshouillers and H. Iwaniec, Kloosterman sums and Fourier coefficients of cusp forms, Invent. Math. 70 (1982/83), 219--288.
  • W. Duke, J. B. Friedlander, and H. Iwaniec, Bounds for automorphic $L$-functions, II, Invent. Math. 115 (1994), 219--239., ; Erratum, Invent. Math. 140 (2000), 227--242.
  • --. --. --. --., A quadratic divisor problem, Invent. Math. 115 (1994), 209--217.
  • W. Duke and H. Iwaniec, Bilinear forms in the Fourier coefficients of half-integral weight cusp forms and sums over primes, Math. Ann. 286 (1990), 783--802.
  • W. Duke and E. Kowalski, A problem of Linnik for elliptic curves and mean-value estimates for automorphic representations, Invent. Math. 139 (2000), 1--39.
  • B. Edixhoven, Rational torsion points on elliptic curves over number fields (after Kamienny and Mazur), Astérisque 227 (1995), 209--227., Séminaire Bourbaki 1993/94, exp. no. 782.
  • A. Erdélyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Higher Transcendental Functions, Vol. II, McGraw-Hill, New York, 1953.
  • --------, Tables of Integral Transforms, Vol. I, II, McGraw-Hill, New York, 1954.,
  • B. H. Gross, ``Heights and the special values of $L$-series'' in Number Theory (Montreal, 1985), CMS Conf. Proc. 7, Amer. Math. Soc., Providence, 1987, 115--187.
  • B. Gross, W. Kohnen, and D. Zagier, Heegner points and derivatives of $L$-series, II, Math. Ann. 278 (1987), 497--562.
  • B. H. Gross and D. B. Zagier, Heegner points and derivatives of $L$-series, Invent. Math. 84 (1986), 225--320.
  • 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.
  • H. Iwaniec, Introduction to the Spectral Theory of Automorphic Forms, Bibl. Rev. Mat. Iberoamericana, Rev. Mat. Iberoamericana, Madrid, 1995.
  • --------, Topics in Classical Automorphic Forms, Grad. Stud. Math. 17, Amer. Math. Soc., Providence, 1997.
  • 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. \CMP1 828 743
  • H. Iwaniec and P. Sarnak, The non-vanishing of central values of automorphic $L$-functions and Landau-Siegel zeros, Israel J. Math. 120, part A (2000), 155--177.
  • --. --. --. --., ``Perspectives on the analytic theory of $L$-functions'' in GAFA 2000: Visions in Mathematics, Towards 2000 (Tel Aviv, 1999), Geom. Funct. Anal. 2000, special volume, part 2, Birkhäuser, Basel, 2000, 705--741.
  • M. Jutila, Lectures on a Method in the Theory of Exponential Sums, Tata Inst. Fund. Res. Lectures on Math. and Phys. 80, Springer, Berlin, 1987.
  • 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.
  • --. --. --. --., A lower bound for the rank of $J_0(q)$, Acta Arith. 94 (2000), 303--343. \CMP1 779 946
  • E. Kowalski, P. Michel, and J. VanderKam, Mollification of the fourth moment of automorphic $L$-functions and arithmetic applications, Invent. Math. 142 (2000), 95--151.
  • --. --. --. --., Non-vanishing of high derivatives of automorphic $L$-functions at the center of the critical strip, J. Reine Angew. Math. 526 (2000), 1--34.
  • W. C. W. Li, Newforms and functional equations, Math. Ann. 212 (1975), 285--315.
  • --. --. --. --., $L$-series of Rankin type and their functional equations, Math. Ann. 244 (1979), 135--166.
  • T. Meurman, On exponential sums involving the Fourier coefficients of Maass wave forms, J. Reine Angew. Math. 384 (1988), 192--207.
  • P. Michel, Complement to ``Rankin-Selberg $L$ functions in the level aspect,'' unpublished notes, 2000, http://gauss.math.univ-montp2.fr/~michel/publi.html
  • P. Michel and J. M. Vanderkam, Triple non-vanishing of twists of automorphic $L$-functions, to appear in Compositio Math.
  • C. J. Moreno, ``Analytic properties of Euler products of automorphic representations'' in Modular Functions of One Variable (Bonn, 1976), VI, Lecture Notes in Math. 627, Springer, Berlin, 1977, 11--26.
  • M. Ram Murty [Murty, M. Ram], ``The analytic rank of $J\sb 0(N)({Q})$'' in Number Theory (Halifax, N.S., 1994), CMS Conf. Proc. 15, Amer. Math. Soc., Providence, 1995, 263--277.
  • D. Ramakrishnan, Modularity of the Rankin-Selberg $L$-series, and multiplicity one for $\SL(2)$, Ann. of Math. (2) 152 (2000), 45--111.
  • K. A. Ribet, ``Galois representations attached to eigenforms with Nebentypus'' in Modular Functions of One Variable (Bonn, 1976), V, Lecture Notes in Math. 601, Springer, Berlin, 1977, 17--51.
  • D. E. Rohrlich, ``The vanishing of certain Rankin-Selberg convolutions'' in Automorphic Forms and Analytic Number Theory (Montreal, 1989), Univ. Montreal, Montreal, 1990, 123--133.
  • K. Rubin, ``Euler systems and modular elliptic curves'' in Galois Representations in Arithmetic Algebraic Geometry (Durham, 1996), London Math. Soc. Lecture Note Ser. 254, Cambridge Univ. Press, Cambridge, 1998, 351--367.
  • Z. Rudnick and P. Sarnak, The behaviour of eigenstates of arithmetic hyperbolic manifolds, Comm. Math. Phys. 161 (1994), 195--213.
  • P. Sarnak, Estimates for Rankin-Selberg $L$-functions and quantum unique ergodicity, J. Funct. Anal. 184 (2001), 419--453. \CMP1 851 004
  • A. J. Scholl, ``An introduction to Kato's Euler systems'' in Galois Representations in Arithmetic Algebraic Geometry (Durham, 1996), London Math. Soc. Lecture Note Ser. 254, Cambridge Univ. Press, Cambridge, 1998, 379--460.
  • F. Shahidi, ``Best estimates for Fourier coefficients of Maass forms'' in Automorphic Forms and Analytic Number Theory (Montreal, 1989), Univ. Montreal, Montreal, 1990, 135--141.
  • J. M. VanderKam, The rank of quotients of $J\sb 0(N)$, Duke Math. J. 97 (1999), 545--577.
  • --. --. --. --., Linear independence of Hecke operators in the homology of $X_0(N)$, J. London Math. Soc. (2) 61 (2000), 349--358.
  • T. Watson, Rankin triple products and quantum chaos, Ph.D. dissertation, Princeton University, Princeton, 2002, http://math.ucla.edu/~tcwatson/thesis
  • S. Zhang, Heights of Heegner cycles and derivatives of $L$-series, Invent. Math. 130 (1997), 99--152.
  • --. --. --. --., Heights of Heegner points on Shimura curves, Ann. of Math. (2) 153 (2001), 27--147. \CMP1 826 411