|
A density theorem on automorphic -functions and some applications
Authors:
Yuk-Kam Lau and Jie Wu
Journal:
Trans. Amer. Math. Soc. 358 (2006), 441-472
MSC (2000):
Primary 11F67, 11F30
Posted:
August 1, 2005
MathSciNet review:
2171241
Full-text PDF Free Access
Abstract |
References |
Similar Articles |
Additional Information
Abstract: We establish a density theorem on automorphic -functions and give some applications on the extreme values of these -functions at and the distribution of the Hecke eigenvalue of holomorphic cusp forms.
References
- 1.
J. Cogdell and P. Michel, On the complex moments of symmetric power
𝐿-functions at 𝑠=1, Int. Math. Res. Not. , posted on
(2004), no. 31, 1561–1617. MR 2035301
(2005f:11094), http://dx.doi.org/10.1155/S1073792804132455
- 2.
S. Gelbart & H. Jacquet, A relation between automorphic representations of GL(2) and GL(3), Ann. Sci. École Norm. Sup. (4) 11 (1978), 471-552. MR 0533066 (81e:10025)
- 3.
Jeffrey Hoffstein and Paul Lockhart, Coefficients of Maass forms and
the Siegel zero, Ann. of Math. (2) 140 (1994),
no. 1, 161–181. With an appendix by Dorian Goldfeld, Hoffstein
and Daniel Lieman. MR 1289494
(95m:11048), http://dx.doi.org/10.2307/2118543
- 4.
Andrew Granville and K. Soundararajan, Large character sums, J. Amer. Math. Soc. 14 (2001), no. 2, 365–397 (electronic). MR 1815216
(2002h:11074), /jams/2001-14-02/S0894-0347-00-00357-X/
- 5.
L. Habsieger & E. Royer,
-functions of automorphic forms and combinatorics: Dyck paths, Ann. Inst. Fourier (Grenoble), to appear.
- 6.
Jeffrey Hoffstein and Paul Lockhart, Coefficients of Maass forms and
the Siegel zero, Ann. of Math. (2) 140 (1994),
no. 1, 161–181. With an appendix by Dorian Goldfeld, Hoffstein
and Daniel Lieman. MR 1289494
(95m:11048), http://dx.doi.org/10.2307/2118543
- 7.
H. Iwaniec, Topics in Classical Automorphic Forms, Graduate Studies in Mathematics, vol. 17, American Mathematical Society, Providence, Rhode Island, 1997. MR 1474964 (98e:11051)
- 8.
H. Iwaniec, W. Luo & P. Sarnak, Low lying zeros of families of
-functions, Inst. Hautes Études Sci. Publ. Math. 91 (2000), 55-131. MR 1828743 (2002h:11081)
- 9.
Henry H. Kim, Functoriality for the exterior square
of 𝐺𝐿₄ and the symmetric fourth of
𝐺𝐿₂, 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 (2003k:11083), /jams/2003-16-01/S0894-0347-02-00410-1/
- 10.
Henry H. Kim and Freydoon Shahidi, Functorial products for
𝐺𝐿₂×𝐺𝐿₃ and the
symmetric cube for 𝐺𝐿₂, Ann. of Math. (2)
155 (2002), no. 3, 837–893. With an appendix by
Colin J. Bushnell and Guy Henniart. MR 1923967
(2003m:11075), http://dx.doi.org/10.2307/3062134
- 11.
Henry H. Kim and Freydoon Shahidi, Cuspidality of symmetric powers with
applications, Duke Math. J. 112 (2002), no. 1,
177–197. MR 1890650
(2003a:11057), http://dx.doi.org/10.1215/S0012-9074-02-11215-0
- 12.
A.W. Knapp, Local Langlands correspondence: the Archimedean case, Proc. Sympos. Pure Math. 55 (1994), Part 2, 393-410. MR 1265560 (95d:11066)
- 13.
E. Kowalski and P. Michel, Zeros of families of automorphic
𝐿-functions close to 1, Pacific J. Math. 207
(2002), no. 2, 411–431. MR 1972253
(2004e:11047), http://dx.doi.org/10.2140/pjm.2002.207.411
- 14.
D.H. Lehmer, Some functions of Ramanujan, Math. Student 27 (1959), 105-116. MR 0131412 (24:A1263)
- 15.
Wenzhi Luo, Values of symmetric square 𝐿-functions at 1,
J. Reine Angew. Math. 506 (1999), 215–235. MR 1665705
(2001d:11055), http://dx.doi.org/10.1515/crll.1999.007
- 16.
H.L. Montgomery, Topics in multiplicative number theory, Lecture Notes in Mathematics 227, Springer-Verlag, Bermion-Heidelberg-New York, 1971. MR 0337847 (49:2616)
- 17.
R.A. Rankin, An
-result for the coefficients of cusp forms, Math. Ann. 283 (1973), 239-250. MR 0321876 (48:241)
- 18.
Emmanuel Royer, Statistique de la variable aléatoire
𝐿(𝑠𝑦𝑚²𝑓,1), Math. Ann.
321 (2001), no. 3, 667–687 (French, with
English and French summaries). MR 1871974
(2003c:11045), http://dx.doi.org/10.1007/s002080100244
- 19.
Emmanuel 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 (French, with English and French
summaries). MR
2013928 (2004k:11078), http://dx.doi.org/10.1016/S0012-9593(03)00024-7
- 20.
E. Royer & J. Wu, Taille des valeurs de fonctions
de carrés symétriques au bord de la bande critique, Revista Matemática Iberoamericana 21 (2005), 263-312.
- 21.
E. Royer & J. Wu, Central values, values at the edge of the critical strip of symmetric power
-functions and Hecke eigenvalues, preprint.
- 22.
Zeév Rudnick and Peter Sarnak, Zeros of principal
𝐿-functions and random matrix theory, Duke Math. J.
81 (1996), no. 2, 269–322. A celebration of
John F. Nash, Jr. MR 1395406
(97f:11074), http://dx.doi.org/10.1215/S0012-7094-96-08115-6
- 23.
J.-P. Serre, Quelques applications du théorème de densité de Chebotarev, Inst. Hautes Études Sci. Publ. Math. 54 (1981), 323-401. MR 0644559 (83k:12011)
- 24.
Jean-Pierre Serre, Répartition asymptotique des
valeurs propres de l’opérateur de Hecke
𝑇_{𝑝}, J. Amer. Math. Soc.
10 (1997), no. 1,
75–102 (French). MR 1396897
(97h:11048), /jams/1997-10-01/S0894-0347-97-00220-8/
- 25.
Gérald Tenenbaum and Jie Wu, Moyennes de certaines fonctions
multiplicatives sur les entiers friables, J. Reine Angew. Math.
564 (2003), 119–166 (French, with English summary).
MR
2021037 (2004m:11151), http://dx.doi.org/10.1515/crll.2003.087
- 26.
E.C. Titchmarsh, The theory of function, Second edition, Oxford University Press, Oxford, 1952. MR 0197687 (33:5850)
- 27.
E.C. Titchmarsh, The theory of the Riemann zeta-function, 2nd edition revised by D.R. Heath-Brown, Oxford. MR 0882550 (88c:11049)
Similar Articles
Retrieve articles in Transactions of the American Mathematical Society
with MSC (2000):
11F67,
11F30
Retrieve articles in all journals
with MSC (2000):
11F67,
11F30
Additional Information
Yuk-Kam Lau
Affiliation:
Department of Mathematics, The University of Hong Kong, Pokfulam Road, Hong Kong
Email:
yklau@maths.hku.hk
Jie Wu
Affiliation:
Institut Elie Cartan, UMR 7502 UHP-CNRS-INRIA, Université Henri Poincaré, 54506 Vandouvre-lès-Nancy, France
Email:
wujie@iecn.u-nancy.fr
DOI:
http://dx.doi.org/10.1090/S0002-9947-05-03774-8
PII:
S 0002-9947(05)03774-8
Keywords:
Special values of automorphic $L$-series,
Fourier coefficients of automorphic forms
Received by editor(s):
November 8, 2003
Received by editor(s) in revised form:
June 23, 2004
Posted:
August 1, 2005
Article copyright:
© Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain after
28 years from publication.
|