Special values of symmetric power L -functions and Hecke eigenvalues

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

Journal de Théorie des Nombres de Bordeaux (2007)

  • Volume: 19, Issue: 3, page 703-753
  • ISSN: 1246-7405

Abstract

top
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.

How to cite

top

Royer, Emmanuel, and Wu, Jie. "Special values of symmetric power $L$-functions and Hecke eigenvalues." Journal de Théorie des Nombres de Bordeaux 19.3 (2007): 703-753. <http://eudml.org/doc/249977>.

@article{Royer2007,
abstract = {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.},
affiliation = {Laboratoire de mathématiques UMR6620 UBP CNRS Université Blaise Pascal Campus universitaire des Cézeaux F–63177 Aubière Cedex, France; 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},
author = {Royer, Emmanuel, Wu, Jie},
journal = {Journal de Théorie des Nombres de Bordeaux},
keywords = {moments of -functions of symmetric powers of modular forms; edge of critical strip; twisted by central value of -functions of modular forms; distribution of small and large Hecke eigenvalues; simultaneous extremality conditions},
language = {eng},
number = {3},
pages = {703-753},
publisher = {Université Bordeaux 1},
title = {Special values of symmetric power $L$-functions and Hecke eigenvalues},
url = {http://eudml.org/doc/249977},
volume = {19},
year = {2007},
}

TY - JOUR
AU - Royer, Emmanuel
AU - Wu, Jie
TI - Special values of symmetric power $L$-functions and Hecke eigenvalues
JO - Journal de Théorie des Nombres de Bordeaux
PY - 2007
PB - Université Bordeaux 1
VL - 19
IS - 3
SP - 703
EP - 753
AB - 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.
LA - eng
KW - moments of -functions of symmetric powers of modular forms; edge of critical strip; twisted by central value of -functions of modular forms; distribution of small and large Hecke eigenvalues; simultaneous extremality conditions
UR - http://eudml.org/doc/249977
ER -

References

top
  1. 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. Zbl1090.11055MR1993332
  2. 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. Zbl1075.11058MR2149530
  3. 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. Zbl1093.11032MR2035301
  4. M. Eichler, Quaternäre quadratische Formen und die Riemannsche Vermutung. Archiv der Mathematik V (1954), 355–366. Zbl0059.03804MR63406
  5. P. D. T. A. Elliott, On the distribution of the values of quadratic L -series in the half-plane σ &gt; 1 2 . Invent. Math. 21 (1973), 319–338. Zbl0265.10022MR352019
  6. S. Friedberg and J. Hoffstein, Nonvanishing theorems for automorphic L -functions on GL ( 2 ) . Ann. of Math. (2) 142 (1995), no. 2, 385–423. Zbl0847.11026MR1343325
  7. 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). Zbl1064.11038MR2114776
  8. 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]. Zbl0814.11032MR1289494
  9. 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. Zbl0406.10022MR533066
  10. A. Granville and K. Soundararajan, Large character sums. J. Amer. Math. Soc. 14 (2001), no. 2, 365–397 (electronic). Zbl0983.11053MR1815216
  11. A. Granville and K. Soundararajan, The distribution of values of L ( 1 , χ d ) . Geom. Funct. Anal. 13 (2003), no. 5, 992–1028. Zbl1044.11080MR2024414
  12. 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. Zbl0861.11032MR1388847
  13. 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. Zbl0814.11032MR1289494
  14. J. Hoffstein and D. Ramakrishnan, Siegel zeros and cusp forms. Internat. Math. Res. Notices (1995), no. 6, 279–308. Zbl0847.11043MR1344349
  15. 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). Zbl1131.11340MR2139690
  16. Jun-ichi Igusa, Kroneckerian model of fields of elliptic modular functions. Amer. J. Math. 81 (1959), 561–577. Zbl0093.04502MR108498
  17. H. Iwaniec and E. Kowalski, Analytic number theory. American Mathematical Society Colloquium Publications, vol. 53, American Mathematical Society, Providence, RI, 2004. Zbl1059.11001MR2061214
  18. 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). Zbl1012.11041MR1828743
  19. 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. Zbl0992.11037MR1815374
  20. 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. Zbl1018.11024MR1937203
  21. 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. Zbl1040.11036MR1923967
  22. H.H. Kim and F. Shahidi, Cuspidality of symmetric powers with applications. Duke Math. J. 112 (2002), no. 1, 177–197. Zbl1074.11027MR1890650
  23. 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. Zbl1054.11026MR1784797
  24. J.E. Littlewood, On the class number of the corpus P ( - k ) . Proc. London Math. Soc. 27 (1928), 358–372. Zbl54.0206.02
  25. W. Luo, Values of symmetric square L -functions at 1 . J. Reine Angew. Math. 506 (1999), 215–235. Zbl0969.11018MR1665705
  26. W. Luo, Nonvanishing of L -values and the Weyl law. Ann. of Math. (2) 154 (2001), no. 2, 477–502. Zbl1003.11019MR1865978
  27. 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). Zbl1078.11032MR2171241
  28. Y.-K. Lau and J. Wu, A large sieve inequality of Elliott-Montgomery-Vaughan type for automorphic forms and two applications. preprint (2007). 
  29. 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. Zbl1168.11016MR2331346
  30. 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. Zbl0942.11040MR1689558
  31. E. Royer, Statistique de la variable aléatoire L ( sym 2 f , 1 ) . Math. Ann. 321 (2001), no. 3, 667–687. Zbl1006.11023MR1871974
  32. 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. Zbl1050.11055MR2013928
  33. D. Ramakrishnan and S. Wang, On the exceptional zeros of Rankin-Selberg L -functions. Compositio Math. 135 (2003), no. 2, 211–244. Zbl1043.11046MR1955318
  34. 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. Zbl1147.11027MR2155022
  35. 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. Zbl0628.10028MR1018385
  36. 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. Zbl0871.11032
  37. G. Tenenbaum and J. Wu, Moyennes de certaines fonctions multiplicatives sur les entiers friables. J. Reine Angew. Math. 564 (2003), 119–166. Zbl1195.11132MR2021037
  38. 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. Zbl0172.18404MR229863

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.