Special values of symmetric power L-functions and Hecke eigenvalues
Journal de Théorie des Nombres de Bordeaux, Volume 19 (2007) no. 3, pp. 703-753.

We compute the moments of L-functions of symmetric powers of modular forms at the edge of the critical strip, twisted by the central value of the L-functions of modular forms. We show that, in the case of even powers, it is equivalent to twist by the value at the edge of the critical strip of the symmetric square L-functions. We deduce information on the size of symmetric power L-functions at the edge of the critical strip in subfamilies. In a second part, we study the distribution of small and large Hecke eigenvalues. We deduce information on the simultaneous extremality conditions on the values of L-functions of symmetric powers of modular forms at the edge of the critical strip.

On calcule les moments des fonctions L de puissances symétriques de formes modulaires au bord de la bande critique en les tordant par les valeurs centrales des fonctions L de formes modulaires. Dans le cas des puissances paires, on montre qu’il est équivalent de tordre par la valeur au bord des fonctions L de carrés symétriques. On en déduit des informations sur la taille des valeurs au bord de la bande critique de fonctions L de puissances symétriques dans certaines sous-familles. Dans une deuxième partie, on étudie la répartition des petites et grandes valeurs propres de Hecke. On en déduit des informations sur des conditions d’extrémalité simultanées des valeurs de fonctions L de puissances symétriques au bord de la bande critique.

Received:
Published online:
DOI: 10.5802/jtnb.609
Emmanuel Royer 1; Jie Wu 2

1 Laboratoire de mathématiques UMR6620 UBP CNRS Université Blaise Pascal Campus universitaire des Cézeaux F–63177 Aubière Cedex, France
2 Institut Élie Cartan UMR7502 UHP CNRS INRIA, Université Henri Poincaré, Nancy 1 F–54506 Vandœuvre-lés-Nancy, France School of Mathematical Sciences Shandong Normal University Jinan, Shandong, 250014, P.R. of China
@article{JTNB_2007__19_3_703_0,
     author = {Emmanuel Royer and Jie Wu},
     title = {Special values of symmetric power $L$-functions and {Hecke} eigenvalues},
     journal = {Journal de Th\'eorie des Nombres de Bordeaux},
     pages = {703--753},
     publisher = {Universit\'e Bordeaux 1},
     volume = {19},
     number = {3},
     year = {2007},
     doi = {10.5802/jtnb.609},
     zbl = {1196.11071},
     mrnumber = {2388795},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.609/}
}
TY  - JOUR
TI  - Special values of symmetric power $L$-functions and Hecke eigenvalues
JO  - Journal de Théorie des Nombres de Bordeaux
PY  - 2007
DA  - 2007///
SP  - 703
EP  - 753
VL  - 19
IS  - 3
PB  - Université Bordeaux 1
UR  - https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.609/
UR  - https://zbmath.org/?q=an%3A1196.11071
UR  - https://www.ams.org/mathscinet-getitem?mr=2388795
UR  - https://doi.org/10.5802/jtnb.609
DO  - 10.5802/jtnb.609
LA  - en
ID  - JTNB_2007__19_3_703_0
ER  - 
%0 Journal Article
%T Special values of symmetric power $L$-functions and Hecke eigenvalues
%J Journal de Théorie des Nombres de Bordeaux
%D 2007
%P 703-753
%V 19
%N 3
%I Université Bordeaux 1
%U https://doi.org/10.5802/jtnb.609
%R 10.5802/jtnb.609
%G en
%F JTNB_2007__19_3_703_0
Emmanuel Royer; Jie Wu. Special values of symmetric power $L$-functions and Hecke eigenvalues. Journal de Théorie des Nombres de Bordeaux, Volume 19 (2007) no. 3, pp. 703-753. doi : 10.5802/jtnb.609. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.609/

[CFKRS03] J. B. Conrey, D. W. Farmer, J. P. Keating, M. O. Rubinstein, and N. C. Snaith, Autocorrelation of random matrix polynomials. Comm. Math. Phys. 237 (2003), no. 3, 365–395. | MR: 1993332 | Zbl: 1090.11055

[CFKRS05] 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), no. 1, 33–104. | MR: 2149530 | Zbl: 1075.11058

[CM04] J. Cogdell and P. Michel, On the complex moments of symmetric power L-functions at s=1. Int. Math. Res. Not. (2004), no. 31, 1561–1617. | MR: 2035301 | Zbl: 1093.11032

[Eic54] M. Eichler, Quaternäre quadratische Formen und die Riemannsche Vermutung. Archiv der Mathematik V (1954), 355–366. | MR: 63406 | Zbl: 0059.03804

[Ell73] P. D. T. A. Elliott, On the distribution of the values of quadratic L-series in the half-plane σ>1 2. Invent. Math. 21 (1973), 319–338. | MR: 352019 | Zbl: 0265.10022

[FH95] S. Friedberg and J. Hoffstein, Nonvanishing theorems for automorphic L-functions on GL (2). Ann. of Math. (2) 142 (1995), no. 2, 385–423. | MR: 1343325 | Zbl: 0847.11026

[FOP04] S. Frechette, K. Ono, and M. Papanikolas, Combinatorics of traces of Hecke operators. Proc. Natl. Acad. Sci. USA 101 (2004), no. 49, 17016–17020 (electronic). | MR: 2114776 | Zbl: 1064.11038

[GHL94] D. Goldfeld, J. Hoffstein, and D. Lieman, An effective zero-free region. Ann. of Math. (2) 140 (1994), no. 1, 177–181, Appendix of [HL94]. | MR: 1289494 | Zbl: 0814.11032

[GJ78] S. Gelbart and H. Jacquet, A relation between automorphic representations of GL (2) and GL (3). Ann. Sci. École Norm. Sup. (4) 11 (1978), no. 4, 471–542. | Numdam | MR: 533066 | Zbl: 0406.10022

[GS01] A. Granville and K. Soundararajan, Large character sums. J. Amer. Math. Soc. 14 (2001), no. 2, 365–397 (electronic). | MR: 1815216 | Zbl: 0983.11053

[GS03] A. Granville and K. Soundararajan, The distribution of values of L(1,χ d ). Geom. Funct. Anal. 13 (2003), no. 5, 992–1028. | MR: 2024414 | Zbl: 1044.11080

[Guo96] J. Guo, On the positivity of the central critical values of automorphic L-functions for GL (2). Duke Math. J. 83 (1996), no. 1, 157–190. | MR: 1388847 | Zbl: 0861.11032

[HL94] J. Hoffstein and P. Lockhart, Coefficients of Maass forms and the Siegel zero. Ann. of Math. (2) 140 (1994), no. 1, 161–181, With an appendix by D. Goldfeld, J. Hoffstein and D. Lieman. | MR: 1289494 | Zbl: 0814.11032

[HR95] J. Hoffstein and D. Ramakrishnan, Siegel zeros and cusp forms. Internat. Math. Res. Notices (1995), no. 6, 279–308. | MR: 1344349 | Zbl: 0847.11043

[HR04] L. Habsieger and E. Royer, L-functions of automorphic forms and combinatorics: Dyck paths. Ann. Inst. Fourier (Grenoble) 54 (2004), no. 7, 2105–2141 (2005). | Numdam | MR: 2139690 | Zbl: pre02162451

[Igu59] Jun-ichi Igusa, Kroneckerian model of fields of elliptic modular functions. Amer. J. Math. 81 (1959), 561–577. | MR: 108498 | Zbl: 0093.04502

[IK04] H. Iwaniec and E. Kowalski, Analytic number theory. American Mathematical Society Colloquium Publications, vol. 53, American Mathematical Society, Providence, RI, 2004. | MR: 2061214 | Zbl: 1059.11001

[ILS00] H. Iwaniec, W. Luo, and P. Sarnak, Low lying zeros of families of L-functions. Inst. Hautes Études Sci. Publ. Math. (2000), no. 91, 55–131 (2001). | Numdam | MR: 1828743 | Zbl: 1012.11041

[IS00] H. Iwaniec and 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: 1815374 | Zbl: 0992.11037

[Kim03] H. H. Kim, Functoriality for the exterior square of GL 4 and the symmetric fourth of GL 2 . J. Amer. Math. Soc. 16 (2003), no. 1, 139–183 (electronic), With appendix 1 by Dinakar Ramakrishnan and appendix 2 by Kim and Peter Sarnak. | MR: 1937203 | Zbl: 1018.11024

[KMV00] E. Kowalski, P. Michel, and J. VanderKam, Mollification of the fourth moment of automorphic L-functions and arithmetic applications. Invent. Math. 142 (2000), no. 1, 95–151. | MR: 1784797 | Zbl: 1054.11026

[KS02a] H. H. Kim and F. Shahidi, Functorial products for GL 2 × GL 3 and the symmetric cube for GL 2 . Ann. of Math. (2) 155 (2002), no. 3, 837–893, With an appendix by Colin J. Bushnell and Guy Henniart. | MR: 1923967 | Zbl: 1040.11036

[KS02b] H.H. Kim and F. Shahidi, Cuspidality of symmetric powers with applications. Duke Math. J. 112 (2002), no. 1, 177–197. | MR: 1890650 | Zbl: 1074.11027

[Lit28] J.E. Littlewood, On the class number of the corpus P(-k). Proc. London Math. Soc. 27 (1928), 358–372.

[Luo99] W. Luo, Values of symmetric square L-functions at 1. J. Reine Angew. Math. 506 (1999), 215–235. | MR: 1665705 | Zbl: 0969.11018

[Luo01] W. Luo, Nonvanishing of L-values and the Weyl law. Ann. of Math. (2) 154 (2001), no. 2, 477–502. | MR: 1865978 | Zbl: 1003.11019

[LW06] Y.-K. Lau and J. Wu, A density theorem on automorphic L-functions and some applications. Trans. Amer. Math. Soc. 358 (2006), no. 1, 441–472 (electronic). | MR: 2171241 | Zbl: 1078.11032

[LW07] Y.-K. Lau and J. Wu, A large sieve inequality of Elliott-Montgomery-Vaughan type for automorphic forms and two applications. preprint (2007).

[Mic02] P. Michel, Analytic number theory and families of automorphic L-functions. in Automorphic Forms and Applications. P. Sarnak & F. Shahidi ed., IAS/Park City Mathematics Series, vol. 12, American Mathematical Society, Providence, RI, 2007. | MR: 2331346 | Zbl: pre05233939

[MV99] H. L. Montgomery and R. C. Vaughan, Extreme values of Dirichlet L-functions at 1. Number theory in progress, Vol. 2 (Zakopane-Kościelisko, 1997), de Gruyter, Berlin, 1999, pp. 1039–1052. | MR: 1689558 | Zbl: 0942.11040

[Roy01] E. Royer, Statistique de la variable aléatoire L( sym 2 f,1). Math. Ann. 321 (2001), no. 3, 667–687. | MR: 1871974 | Zbl: 1006.11023

[Roy03] E. Royer, Interprétation combinatoire des moments négatifs des valeurs de fonctions L au bord de la bande critique. Ann. Sci. École Norm. Sup. (4) 36 (2003), no. 4, 601–620. | Numdam | MR: 2013928 | Zbl: 1050.11055

[RW03] D. Ramakrishnan and S. Wang, On the exceptional zeros of Rankin-Selberg L-functions. Compositio Math. 135 (2003), no. 2, 211–244. | MR: 1955318 | Zbl: 1043.11046

[RW05] E. Royer and J. Wu, Taille des valeurs de fonctions L de carrés symétriques au bord de la bande critique. Rev. Mat. Iberoamericana 21 (2005), no. 1, 263–312. | MR: 2155022 | Zbl: pre02212054

[Sar87] P. Sarnak, Statistical properties of eigenvalues of the Hecke operators. Analytic number theory and Diophantine problems (Stillwater, OK, 1984), Birkhäuser Boston, Boston, MA, 1987, pp. 321–331. | MR: 1018385 | Zbl: 0628.10028

[Ser97] J.-P. Serre, Répartition asymptotique des valeurs propres de l’opérateur de Hecke T p . J. Amer. Math. Soc. 10 (1997), no. 1, 75–102. | Zbl: 0871.11032

[TW03] G. Tenenbaum and J. Wu, Moyennes de certaines fonctions multiplicatives sur les entiers friables. J. Reine Angew. Math. 564 (2003), 119–166. | MR: 2021037 | Zbl: pre02005601

[Vil68] N. Ja. Vilenkin, Special functions and the theory of group representations. Translated from the Russian by V. N. Singh. Translations of Mathematical Monographs, Vol. 22, American Mathematical Society, Providence, R. I., 1968. | MR: 229863 | Zbl: 0172.18404

Cited by Sources: