A new method for lower bounds of <i>L</i>-functions

Stephen S. Gelbart; Erez M. Lapid; Peter Sarnak

doi:10.1016/j.crma.2004.04.024

Number Theory

A new method for lower bounds of L-functions
[Une nouvelle méthode pour minorer des fonctions L.]

Présenté par : Jean-Pierre Serre

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

Comptes Rendus. Mathématique, Volume 339 (2004) no. 2, pp. 91-94.

Résumé
Abstract

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 $R (s) > 1$ .

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 $R (s) > 1$ .

Reçu le : 2004-04-17
Accepté le : 2004-04-27
Publié le : 2004-07-15

DOI : 10.1016/j.crma.2004.04.024

Affiliations des auteurs :

Stephen S. Gelbart ¹ ; Erez M. Lapid ² ; Peter Sarnak ^{3, 4}

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     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},
     pages = {91--94},
     publisher = {Elsevier},
     volume = {339},
     number = {2},
     year = {2004},
     doi = {10.1016/j.crma.2004.04.024},
     language = {en},
}

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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. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2004.04.024/

Bibliographie
Cité par

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

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[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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