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On the Exceptional Zeros of Rankin–Selberg L-Functions

Ramakrishnan, Dinakar and Wang, Song (2003) On the Exceptional Zeros of Rankin–Selberg L-Functions. Compositio Mathematica, 135 (2). pp. 211-244. ISSN 0010-437X. doi:10.1023/A:1021761421232.

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The main objects of study in this article are two classes of Rankin–Selberg L-functions, namely L(s,ƒ×g) and L(s, sym^2(g)× sym^2(g)), where ƒ, g are newforms, holomorphic or of Maass type, on the upper half plane, and sym^2(g) denotes the symmetric square lift of g to GL(3). We prove that in general, i.e., when these L-functions are not divisible by L-functions of quadratic characters (such divisibility happening rarely), they do not admit any LandauSiegel zeros. Such zeros, which are real and close to s=1, are highly mysterious and are not expected to occur. There are corollaries of our result, one of them being a strong lower bound for special value at s=1, which is of interest both geometrically and analytically. One also gets this way a good bound on the norm of sym^2(g).

Item Type:Article
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Wang, Song0000-0003-3116-5038
Additional Information:© 2003 Kluwer Academic Publishers. Received 7 August 2001; accepted in final form: 29 November 2001. We would like to thank D. Bump, W. Duke, H. Jacquet, H. Kim, W. Luo, S. Miller, F. Shahidi and E. Stade for useful conversations and/or correspondence. Clearly this paper depends on the ideas and results of the articles [HL94], [GHLL94], [HRa95], [Ra2000], [KSh2000,1] and [K2000]. The first author would like to thank the NSF for support through the grants DMS-9801328 and DMS-0100372. The subject matter of this paper formed a portion of the first author’s Schur Lecture at the University of Tel Aviv in March 2001, and he would like to thank J. Bernstein and S. Gelbart for inviting him and for their interest.
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Subject Keywords:factorizations; GL(n); holomorphic forms; Landau–Siegel zeros; lower bound; Maass forms; Petersson norm; positive Dirichlet series; Rankin–Selberg L-functions; selfdual forms; spectral normalization; symmetric power liftings; symmetric space.
Issue or Number:2
Classification Code:MSC (2000): 11F67, 11F70
Record Number:CaltechAUTHORS:20111026-134851953
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Official Citation:Dinakar Ramakrishnan and Song Wang (2003). On the Exceptional Zeros of Rankin–Selberg L-Functions. Compositio Mathematica, 135 , pp 211-244 doi:10.1023/A:1021761421232
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:27458
Deposited By: Ruth Sustaita
Deposited On:26 Oct 2011 21:38
Last Modified:09 Nov 2021 16:48

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