Comptes Rendus
Number Theory
A new method for lower bounds of L-functions
Comptes Rendus. Mathématique, Volume 339 (2004) no. 2, pp. 91-94.

Let L(s,π,r) be an L-function which appears in the Langlands–Shahidi theory. We give a lower bound for L(s,π,r) when R(s)=1 using Eisenstein series. This method is applicable even when L(s,π,r) is not known to be absolutely convergent for R(s)>1.

Soit L(s,π,r) une fonction L présente dans la théorie de Langlands–Shahidi. Nous prouvons une minoration de L(s,π,r) quand R(s)=1, en utilisant les séries d'Eisenstein. Cette méthode s'applique même lorsqu'on ne sait pas que L(s,π,r) est absolument convergente pour R(s)>1.

Published online:
DOI: 10.1016/j.crma.2004.04.024
Stephen S. Gelbart 1; Erez M. Lapid 2; Peter Sarnak 3, 4

1 Faculty of Mathematics and Computer Science, Nicki and J. Ira Harris Professorial Chair, The Weizmann Institute of Science, Rehovot 76100, Israel
2 Einstein Institute of Mathematics, The Hebrew University of Jerusalem, Jerusalem 91904, Israel
3 Department of Mathematics, Princeton University, Princeton, NJ, USA
4 The Courant Institute, New York, NY, USA
     author = {Stephen S. Gelbart and Erez M. Lapid and Peter Sarnak},
     title = {A new method for lower bounds of {\protect\emph{L}-functions}},
     journal = {Comptes Rendus. Math\'ematique},
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Stephen S. Gelbart; Erez M. Lapid; Peter Sarnak. A new method for lower bounds of L-functions. Comptes Rendus. Mathématique, Volume 339 (2004) no. 2, pp. 91-94. doi : 10.1016/j.crma.2004.04.024.

[1] F. Brumley Effective multiplicity one for GLm (Preprint) | arXiv

[2] S. Gelbart; F. Shahidi Boundedness of automorphic L-functions in vertical strips, J. Amer. Math. Soc., Volume 14 (2001) no. 1, pp. 79-107 (electronic)

[3] J. Hoffstein; P. Lockhart Coefficients of Maass forms and the Siegel zero, Ann. of Math. (2), Volume 140 (1994) no. 1, pp. 161-181 (With an appendix by Dorian Goldfeld, Hoffstein and Daniel Lieman)

[4] J. Hoffstein; D. Ramakrishnan Siegel zeros and cusp forms, Internat. Math. Res. Notices, Volume 6 (1995), pp. 279-308

[5] H. Iwaniec; P. Sarnak Perspectives on the analytic theory of L-functions, Geom. Funct. Anal. (2000), pp. 705-741 (Special Volume, Part II), GAFA 2000 (Tel Aviv, 1999)

[6] H. Jacquet; J.A. Shalika A non-vanishing theorem for zeta functions of GLn, Invent. Math., Volume 38 (1976/77) no. 1, pp. 1-16

[7] H.H. Kim Functoriality for the exterior square of GL4 and the symmetric fourth of GL2, J. Amer. Math. Soc., Volume 16 (2003) no. 1, pp. 139-183 (electronic). With appendix 1 by Dinakar Ramakrishnan and appendix 2 by Kim and Peter Sarnak

[8] H.H. Kim; F. Shahidi Functorial products for GL2×GL3 and the symmetric cube for GL2, Ann. of Math. (2), Volume 155 (2002) no. 3, pp. 837-893 (With an appendix by Colin J. Bushnell and Guy Henniart)

[9] R.P. Langlands Euler products, Yale Mathematical Monographs, vol. 1, Yale University Press, New Haven, CT, 1971 (A James K. Whittemore Lecture in Mathematics given at Yale University, 1967)

[10] E. Lapid, On the spectral expansion of Jacquet's relative trace formula, Preprint

[11] C. Mœglin; J.-L. Waldspurger Spectral Decomposition and Eisenstein Series, Cambridge University Press, Cambridge, 1995 (Une paraphrase de l'Écriture [A paraphrase of Scripture])

[12] C.J. Moreno Analytic proof of the strong multiplicity one theorem, Amer. J. Math., Volume 107 (1985) no. 1, pp. 163-206

[13] W. Müller On the singularities of residual intertwining operators, Geom. Funct. Anal., Volume 10 (2000) no. 5, pp. 1118-1170

[14] P. Sarnak Non-vanishing of L-functions on R(s)=1, Shalikafest (2003)

[15] F. Shahidi On certain L-functions, Amer. J. Math., Volume 103 (1981) no. 2, pp. 297-355

[16] F. Shahidi On the Ramanujan conjecture and finiteness of poles for certain L-functions, Ann. of Math. (2), Volume 127 (1988) no. 3, pp. 547-584

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