Infinite Dimensional Groups and Automorphic L-Functions

Shahidi, Freydoon

doi:10.4310/pamq.2005.v1.n3.a8

Cited by 11 publications

(5 citation statements)

References 46 publications

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“…The automorphic representation π is cuspidal, as one sees from Remark 5.9 and Proposition 5.11 of [Sh3]. Here are two other proofs of the same assertion, both using more the associated ℓ-adic representation.…”

Section: Here Is a Gentle Strengthening Of Theorem Amentioning

confidence: 72%

Siegel modular forms of genus $2$ attached to elliptic curves

Ramakrishnan

Shahidi

2007

Mathematical Research Letters

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Section: Here Is a Gentle Strengthening Of Theorem Amentioning

confidence: 72%

Siegel modular forms of genus $2$ attached to elliptic curves

Ramakrishnan

Shahidi

2007

Mathematical Research Letters

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“…However, whereas the unipotent radical of P can be equipped with a suitable projective limit structure [11, §3] which makes its arithmetic quotient compact, the unipotent radical for P − is not known to have any such structure. F. Shahidi [35] showed that a naive extension of the Langlands-Shahidi method (involving the constant term for P ) could not capture any nontrivial automorphic L-functions. A. Braverman and D. Kazhdan proposed that the constant term for P − contains the anticipated L-functions L(s, φ, ρ), where φ is a cusp form on M and ρ occurs in the adjoint action of L M .…”

Section: 3mentioning

confidence: 99%

“…F. Shahidi [35] showed that a naive extension of the Langlands-Shahidi method (involving the constant term for P ) could not capture any nontrivial automorphic L-functions. A. Braverman and D. Kazhdan proposed that the constant term for P − contains the anticipated L-functions L(s, φ, ρ), where φ is a cusp form on M and ρ occurs in the adjoint action of L M .…”

Section: 3mentioning

confidence: 99%

Entirety of cuspidal Eisenstein series on loop groups

Garland¹,

Miller²,

Patnaik³

2017

American Journal of Mathematics

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Abstract. In this paper, we prove the entirety of loop group Eisenstein series induced from cusp forms on the underlying finite dimensional group, by demonstrating their absolute convergence on the full complex plane. This is quite in contrast to the finite-dimensional setting, where such series only converge absolutely in a right half plane (and have poles elsewhere coming from L-functions in their constant terms). Our result is the Q-analog of a theorem of A. Braverman and D. Kazhdan from the function field setting [4, Theorem 5.2], who previously showed the analogous Eisenstein series there are finite sums.

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“…It is however not precisely clear if this extension of the theory of Eisenstein series will yield necessarily to new types of L-functions and the focus of the discussion has so far been on series defined on affine groups. In fact in [388] an argument was provided that no new functions will be found, while in [152] a new method for obtaining such functions was devised. This new method relies on an expansion of the series with respect to "lower triangular parabolics", instead of only "upper triangular parabolics".…”

Section: Gravity Motivation: Quantum Cosmological Billiardsmentioning

confidence: 99%

Eisenstein series and automorphic representations

Fleig¹,

Gustafsson²,

Kleinschmidt³

et al. 2015

Preprint

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Note to the readerThis book has grown out of our endeavour to understand the theory of automorphic representations and the structure of Fourier expansions of automorphic forms with a particular emphasis on adelic methods and Eisenstein series. Our intention is also to open a path of communication between mathematicians and physicists, in particular string theorists, interested in these topics. Most of the results presented herein exist already in the literature and we benefitted greatly from [66, 98, 183-185, 297, 389, 405, 406]; our exposition differs, however, at places from the standard one. A few new results and examples are included as well; in particular, we provide many techniques for working out aspects of the Fourier expansion of Eisenstein series. We would be very grateful to learn of any omissions and mistakes that we have made unintentionally.Sections that are more advanced or explore topics beyond the main focus of this treatise are marked with an asterisk. discussions with colleagues both from the mathematics community and from the string theory community. We are especially indebted to

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Infinite Dimensional Groups and Automorphic L-Functions

Cited by 11 publications

References 46 publications

Siegel modular forms of genus $2$ attached to elliptic curves

Siegel modular forms of genus $2$ attached to elliptic curves

Entirety of cuspidal Eisenstein series on loop groups

Eisenstein series and automorphic representations

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