Familles de fonctions $L$ de formes automorphes et applications

Michel, Philippe

Journal de théorie des nombres de Bordeaux, Tome 15 (2003) no. 1, pp. 275-307.

Une notion importante qui a émergé de la théorie analytique des fonctions $L$ ces dernières années, est celle de famille. Par exemple les familles de fonctions $L$ interviennent naturellement dans le modèle probabiliste des matrices aléatoires de Katz/Sarnak qui vise à prédire la répartition des zéros des fonctions $L$. L’analyse des fonctions $L$ en famille intervient également dans la résolution (inconditionnelle) de divers problèmes ayant une signification arithmétique profonde, tel que le problème de montrer la non-annulation de valeur spéciales de fonctions $L$ ou encore celui de borner non-trivialement ces valeurs (le problème de convexité). Dans cet article, nous passons en revue les techniques analytiques mises en jeu pour résoudre ces questions et décrivons plusieurs applications de nature arithmétique.

One of the important concept that has emerged these last years in the analytic theory of $L$ functions, is the concept of families. For instance, families of $L$ functions occur naturally in Katz/Sarnak’s probabilistic model of random matrices whose goal is to predict the distribution of zeros of $L$ functions. The study of $L$ functions within families occurs also in the (unconditional) resolution of several problems having some deep arithmetical meaning : the question of non-vanishing of special values of $L$ functions or the problem of giving non-trivial upper bounds for these special values (the subconvexity problem). In this paper, we review the analytic method involved in solution of some of these problems and give several applications.

[BD1] M. Bertolini, H. Darmon, Heegner points of Mumford-Tate curves. Invent. Math. 126 (1996), 413-456. | MR | Zbl

[BD2] M. Bertolini, H. Darmon, A rigid analytic Gross-Zagier formula and arithmetic applications, with an appendix by Bas Edixhoven. Ann. of Math. 146 (1997), 111-147. | MR | Zbl

[Bu] D.A. Burgess On Character sums and L-series. Proc. London Math. Soc. 12 (1962), 193-206. | MR | Zbl

[Co] J. Cogdell, On sums of three squares. J. Théor. Nombres Bordeaux 15 (2003), 33-44. | Numdam | MR | Zbl

[CPSS] J. Cogdell, I.I. Piatetskii-Shapiro, P. Sarnak. En préparation.

[Cor] C. Cornut, Mazur's conjecture on higher Heegner points. Invent. Math. 148 (2002), 495-523. | MR | Zbl

[CI] B. Conrey, H. Iwaniec, The cubic moment of central values of automorphic L-functions. Annals of Math. 151 (2000), 1175-1216. | MR | Zbl

[CS] J.B. Conrey, K. Soundararajan, Real zeros of quadratic Dirichlet L-functions. Invent. Math. 150 (2002), 1-44. | MR | Zbl

[De1] P. Deligne, La conjecture de Weil I. Publ. Math. IHES 43 (1974), 273-308. | Numdam | MR | Zbl

[De2] P. Deligne, La conjecture de Weil II. Publ. Math. IHES 52 (1981), 313-428. | Numdam | MR | Zbl

[Du] W. Duke, Hyperbolic distribution problems and half-integral weight Maass forms. Invent. Math. 92 (1988), 73-90. | MR | Zbl

[DFI1] W. Duke, J. Friedlander, H. Iwaniec, Bounds for automorphic L-functions. Invent. Math. 112 (1993), 1-8. | MR | Zbl

[DFI2] W. Duke, J. Friedlander, H. Iwaniec, A quadratic divisor problem. Invent. Math. 115 (1994), 209-217. | MR | Zbl

[DFI3] W. Duke, J. Friedlander, H. Iwaniec, Bounds for automorphic L-functions, II. Invent. Math. 115 (1994), 219-239. | MR | Zbl

[DFI4] W. Duke, J. Friedlander, H. Iwaniec, Class group L-functions. Duke Math. J. 79 (1995), 1-56. | MR | Zbl

[DFI5] W. Duke, J. Friedlander, H. Iwaniec, Bilinear forms with Kloosterman fractions. Invent. Math. 128 (1997), 23-43. | MR | Zbl

[DFI6] W. Duke, J. Friedlander, H. Iwaniec, Representations by the determinant and mean values of L-functions. Sieve methods, exponential sums, and their applications in number theory (Cardiff, 1995), 109-115, London Math. Soc. Lecture Note Ser., 237, Cambridge Univ. Press, Cambridge, 1997. | MR | Zbl

[DFI7] W. Duke, J. Friedlander, H. Iwaniec, Bounds for automorphic L-functions, III. Invent. Math. 143 (2001), 221-248. | MR | Zbl

[DFI8] W. Duke, J. Friedlander, H. Iwaniec, The subconvexity problem for Artin L functions. Invent. Math. 149 (2002), 489-577. | MR | Zbl

[FoI] É. Fouvry, H. Iwaniec, A subconvexity bound for Hecke L-functions. Ann. Sci. École Norm. Sup. (4) 34 (2001), 669-683. | Numdam | MR | Zbl

[Fr] J.B. Friedlander, Bounds for L-functions. Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zürich, 1994), 363-373, Birkhäuser, Basel, 1995. | MR | Zbl

[Ga] P. Garrett, Decomposition of Eisenstein series; Rankin triple products. Annals of Math. 125 (1987), 209-235. | MR | Zbl

[GJ] S. Gelbart, H. Jacquet, A relation between automorphic representations of GL(2) and GL(3). Ann. Sci. École Norm. Sup. (4) 11 (1978), 471-542. | Numdam | MR | Zbl

[GoJ] R. Godement, H. Jacquet, Zeta functions of simple algebras. Lecture Notes in Mathematics, Vol. 260, 1972. | MR | Zbl

[G] B. Gross, Heights and the special values of L-series. Number theory (Montreal, Que., 1985), 115-187, CMS Conf. Proc. 7, Amer. Math. Soc., Providence, RI, 1987. | MR | Zbl

[GZ] B. Gross, D. Zagier, Heegner points and derivatives of L-series. Invent. Math. 84 (1986), 225-320. | MR | Zbl

[HK] M. Harris, S. Kudla, the central value of a triple product L fuunction. Annals of Math. 133 (1991), 605-672. | MR | Zbl

[HM] D.R. Heath-Brown, P. Michel, Exponential decays for the frequency of the analytic rank of Automorphic L-functions. Duke Math. Journal 102 (2000), 475-484. | MR | Zbl

[HL] J. Hoffstein, P. Lockhart, Coefficients of Maass forms and the Siegel zero. With an appendix by Dorian Goldfield, Hoffstein and Daniel Lieman. Ann. of Math. (2) 140 (1994), 161-181. | MR | Zbl

[HR] J. Hoffstein, D. Ramakrishnan, Siegel zeros and cusp forms. Internat. Math. Res. Notices 1995, no. 6, 279-308. | MR | Zbl

[Iw1] H. Iwaniec, Fourier coefficients of modular forms of half-integral weight. Invent. Math. 87 (1987), 385-401. | MR | Zbl

[Iw2] H. Iwaniec, The spectral growth of automorphic L functions. J. Reine Angew. Math. 428 (1992), 139-159. | MR | Zbl

[IS1] H. Iwaniec, P. Sarnak, The non-vanishing of central values of automorphic L-functions and Landau-Siegel zeros. Israel J. Math. 120 (2000), part A, 155-177. | MR | Zbl

[IS2] H. Iwaniec, P. Sarnak, Perspectives in the Analytic Theory of L functions. GAFA 2000 (Tel Aviv, 1999). Geom. Funct. Anal. 2000, Special Volume, Part II, 705-741. | MR | Zbl

[Iv] A. Ivić, On sums of Hecke series in short intervals. J. Théor. Nombres Bordeaux 13 (2001), 453-468. | Numdam | MR | Zbl

[JS] H. Jacquet, J.A. Shalika, On Euler products and the classification of automorphic representations. I. Amer. J. Math. 103 (1981), 499-558. | MR | Zbl

[JPPS] H. Jacquet, I.I. Piatetskii-Shapiro, J.A. Shalika, Rankin-Selberg convolutions. Amer. J. Math. 105 (1983), 367-464. | MR | Zbl

[KaSa1] N.M. Katz, P. Sarnak, Random matrices, Frobenius eigenvalues, and monodromy. American Mathematical Society Colloquium Publications, 45. American Mathematical Society, Providence, RI, 1999. | MR | Zbl

[KaSa2] N.M. Katz, P. Sarnak, Zeroes of zeta functions and symmetry. Bull. Amer. Math. Soc. (N.S.) 36 (1999), 1-26. | MR | Zbl

[Ki] H. Kim, Functoriality for the exterior square of GL4 and symmetric fourth of GL2. With appendix 1 by Dinakar Ramakrishnan and appendix 2 by Kim and Peter Sarnak. J. Amer. Math. Soc. 16 (2003), 139-183. | MR | Zbl

[KiSh] H. Kim, F. Shahidi, Cuspidality of symmetric powers with applications. Duke Math. J. 112 (2002), 177-197. | MR | Zbl

[Ko] V. Kolyvagin, Euler Systems, The Grothendieck festschrift. Prog. in Math. Boston, Birkhauser, 1990. | MR | Zbl

[KM1] E. Kowalski, P. Michel, The analytic rank of Jo(q) and zeros of automorphic L-functions. Duke Math. Journal 100 (1999), 503-542. | MR | Zbl

[KM2] E. Kowalski, P. Michel, A lower bound for the rank of J0(q). Acta Arith. 94 (2000), 303-343. | MR | Zbl

[KM3] E. Kowalski, P. Michel, Deux Théorèmes de non-annulation pour les valeurs spéciales de fonctions L. Manuscripta Math. 104 (2001), 1-19. | MR | Zbl

[KM4] E. Kowalski, P. Michel, Appendice à "Sur la nature non cyclotomique des points d'ordre fini des courbes elliptiques" de L. Merel. Duke Math. J. 110 (2001), 110-119. | MR | Zbl

[KMV1] E. Kowalski, P. Michel, J. Vanderkam, Non-vanishing of higher derivatives of automorphic L-functions. J. Reine Angew. Math. 526 (2000), 1-34. | MR | Zbl

[KMV2] E. Kowalski, P. Michel, J. Vanderkam, Mollification of the fourth moment of automorphic L-functions and arithmetic applications. Invent. Math. 142 (2000), 95-151. | MR | Zbl

[KMV3] E. Kowalski, P. Michel, J. Vanderkam, Rankin-Selberg L-functions in the level aspect. Duke Math. J. 114 (2002), 123-191. | MR | Zbl

[La] G. Laumon, Transformation de Fourier, constantes d'équations fonctionnelles et conjecture de Weil. Inst. Hautes Études Sci. Publ. Math. No. 65 (1987), 131-210. | Numdam | MR | Zbl

[Lu] W. Luo, On the nonvanishing of Rankin-Selberg L-functions. Duke Math. J. 69 (1993), 411-425. | MR | Zbl

[Me] L. Merel, Sur la nature non-cyclotomique des points d'ordre fini des courbes elliptiques. Duke Math. J. 110 (2001), 81-119. | MR | Zbl

[Mi1] P. Michel, Répartition des zéros des fonctions L et matrices aléatoires. Exposé Bourbaki 887, Mars 2001. | Numdam | Zbl

[Mi2] P. Michel, the subconvexity problem for Rankin-Selberg L functions with nebentypus and the equidistribution of Heegner points. Annals of Math, à paraître. | Zbl

[MV] P. Michel, J. Vanderkam, Simultaneous non-vanishing of twists of automorphic L-functions. Compositio Math. 134 (2002), 135-191.. | MR | Zbl

[Mo] G. Molteni, Upper and lower bounds at s = 1 for certain Dirichlet series with Euler product. Duke Math. J. 111 (2002), 133-158. | MR | Zbl

[PS] R.S. Phillips, P. Sarnak, On cusp forms for co-finite subgroups of PSL(2, R). Invent. Math. 80 (1985), 339-364. | MR | Zbl

[Ra] D. Ramakrishnan, modularity of the Rankin-Selberg L-series, and multiplicity one for SL(2). Annals of Math. 152 (2000), 45-111. | MR | Zbl

[Ru] Z. Rudnick, Notes on zeros of modular forms, preprint, 1999.

[RS] Z. Rudnick, P. Sarnak, The behaviour of eigenstates of arithmetic hyperbolic manifolds. Comm. Math. Phys. 161 (1994), 195-213. | MR | Zbl

[Ro] D. Rohrlich, Elliptic curves and the Weil-Deligne group. Elliptic curves and related topics, 125-157, CRM Proc. Lecture Notes, 4, Amer. Math. Soc., Providence, RI, 1994. | MR | Zbl

[Sa1] P. Sarnak, Integrals of products of eigenfunctions. Internat. Math. Res. Notices 1994, no. 6, 251-260. | MR | Zbl

[Sa2] P. Sarnak, Estimates for Rankin-Selberg L-functions and Quantum Unique Ergodicity. J. Funct. Anal. 184 (2001), 419-453. | MR | Zbl

[Sc] A. Scholl, An introduction to Kato's Euler systems, Galois representations in arithmetic algebraic geometry (Durham, 1996), 379-460, London Math. Soc. Lecture Note Ser., 254, Cambridge Univ. Press, Cambridge, 1998. | MR | Zbl

[Se] A. Selberg, Collected papers. Vol. I, II, (With a foreword by K. Chandrasekharan). Springer-Verlag, Berlin, 1989. | MR | Zbl

[VI] J. Vanderkam, The rank of quotients of Jo(N). Duke Math. J. 97 (1999), 545-577. | MR | Zbl

[V2] J. Vanderkam, Linear independence of Hecke operators in the homology of X0(N). J. London Math. Soc. 61 (2000), 349-358. | MR | Zbl

[Va] Vatsal, Uniform distribution of Heegner points, preprint, 2001. | MR

[Wa1] J.-L. Waldspurger, Sur les coefficients de Fourier des formes modulaires de poids demi-entier. J. Math. Pures Appl. (9) 60 (1981), 375-484. | MR | Zbl

[Wat] T. Watson, Rankin triple products and quantum chaos. Thesis, Princeton Univ., Princeton, NJ, 2000.

[We] H. Weyl, Zur Abschtzung von ζ(1 + ti). Math. Z. 10 (1921), 88-101. | JFM

[Zh1] S. Zhang, Heights of Heegner points on Shimura curves. Ann. of Math. 153 (2001), 27-147. | MR | Zbl

[Zh2] S. Zhang, Gross-Zagier formula for GL(2). Asian J. Math. 5 (2001), 183-290. | MR | Zbl