Nonvanishing of certain Rankin-Selberg L-functions

نویسنده

  • A. Raghuram
چکیده

In this article we prove that given a holomorphic cusp form f and any point s0 in the complex plane, there is a holomorphic cusp form g such that the Rankin-Selberg L-function L(s, f × g) is non-zero at s0. Résumé: Dans cet article, on prouve le résultat suivant. Etat donné une forme holomorphe cuspidale f et un point quelquonque du plan complexe, il existe une forme holomorphe cuspidale g telle que la fonction L(s, f × g) de Rankin-Selberg n’est pas nulle à s0. The aim of this article is to prove that given a holomorphic cusp form f on the upper half plane h, given any point s0 in the complex plane and given any positive integer l there is a holomorphic cusp form g of weight l+1, which is also an eigenform and a newform and such that the Rankin-Selberg L-function L(s, f × g) is non-zero at s0. One may try to prove such a theorem by averaging. Namely, by choosing a suitable set of ‘possible g’s’ and taking the average of L(s, f × g) over this set and then isolating a dominant term and showing it is non-zero. In some sense the point of this paper is to say that once such an averaging has been done in one context [6] then some generalities from the theory of automorphic forms takes over and gives our nonvanishing theorem ‘almost for free’. The main ingredients in our proof are the notion of base change and automorphic induction for automorphic representations of GL(2) (a general reference for which is [1]) and the main theorem of Rohrlich [6]. After the proof of the main theorem we make various remarks wherein we carefully analyze the choices we make in getting hold of the ‘twist’ g and in particular say what the level of g can be. For instance, if l is even, then it is possible to arrange the level to be a squarefree product of 2 primes relatively prime to N and one of these primes can be essentially arbitrary. We then point out variations of this theorem wherein ∗2000 Mathematics Subject Classification : 11F66 (11F67, 11F70, 22E55).

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تاریخ انتشار 2007