Symmetric Square L-Functions on GL(r)

Bump, Daniel; Ginzburg, David

doi:10.2307/2946548

Cited by 96 publications

(168 citation statements)

References 12 publications

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“…But since sym 3 ðpÞ is a cuspidal automorphic representation of GLð4; A F Þ, this is a consequence (see Theorem 3.3) of the main result of [BuG92]. & Proof of Theorem B 0 (contd.)…”

Section: Proof Of Theorem Bmentioning

confidence: 81%

Untitled

Ramakrishnan

Wang

2003

Compositio Mathematica

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Abstract. The main objects of study in this article are two classes of Rankin-Selberg L-functions, namely Lðs; f Â gÞ and Lðs; sym 2 ðgÞ Â sym 2 ðgÞÞ, where f; g are newforms, holomorphic or of Maass type, on the upper half plane, and sym 2 ðgÞ denotes the symmetric square lift of g to GLð3Þ. 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 ¼ 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 ¼ 1, which is of interest both geometrically and analytically. One also gets this way a good bound on the norm of sym 2 ðgÞ.

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“…But since sym 3 ðpÞ is a cuspidal automorphic representation of GLð4; A F Þ, this is a consequence (see Theorem 3.3) of the main result of [BuG92]. & Proof of Theorem B 0 (contd.)…”

Section: Proof Of Theorem Bmentioning

confidence: 81%

Untitled

Ramakrishnan

Wang

2003

Compositio Mathematica

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“…They play a crucial role in the functional equation and analytic continuation of a number of integral representations of L-functions, e.g. [3,4,8,25]. In this section we describe their distributional counterparts.…”

Section: Mirabolic Eisenstein Series For Gl(n)mentioning

confidence: 99%

Adelization of automorphic distributions and mirabolic Eisenstein series

Miller¹,

Schmid²

2011

Representation Theory and Mathematical Physics

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Abstract. Automorphic representations can be studied in terms of the embeddings of abstract models of representations into spaces of functions on Lie groups that are invariant under discrete subgroups. In this paper we describe an adelic framework to describe them for the group GL(n, R), and provide a detailed analysis of the automorphic distributions associated to the mirabolic Eisenstein series. We give an explicit functional equation for some distributional pairings involving this mirabolic Eisenstein distribution, and the action of intertwining operators.

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“…For the proof, apart from the results on the mean value of the error term for the summatory function of the coefficients of a general Dirichlet series (see [7,10]), we also need some analytic properties of L-functions associated with automorphic forms on GL(3) [22]- [24]. Let {a(n)} and {b(n)} be two sequences of complex numbers, not all zero, and let {λ n } and {µ n } be two sequences of real numbers such that…”

Section: C(y)mentioning

confidence: 99%

“…Analytic continuability and the functional equation for L(s, F × F ), relating s and 1 − s, were studied by several authors; see [22]- [24]. The symmetric square L-function of F is given by the Euler product…”

Section: C(y)mentioning

confidence: 99%