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This article is to understand the critical values of $L$-functions $L(s,Piotimes chi)$ and to establish the relation of the relevant global periods at the critical places. Here $Pi$ is an irreducible regular algebraic cuspidal automorphic representation of $mathrm{GL}_{2n}(mathbb A)$ of symplectic type and $chi$ is a finite order automorphic character of $mathrm{GL}_1(mathbb A)$, with $mathbb A$ is the ring of adeles of a number field $mathrm k$.
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We prove the Archimedean period relations for Rankin-Selberg convolutions for $mathrm{GL}(n)times mathrm{GL}(n-1)$. This implies the period relations for critical values of the Rankin-Selberg L-functions for $mathrm{GL}(n)times mathrm{GL}(n-1)$.
Period polynomials have long been fruitful tools for the study of values of $L$-functions in the context of major outstanding conjectures. In this paper, we survey some facets of this study from the perspective of Eichler cohomology. We discuss ways to incorporate non-cuspidal modular forms and values of derivatives of $L$-functions into the same framework. We further review investigations of the location of zeros of the period polynomial as well as of its analogue for $L$-derivatives.
In this brief essay a construction of the $2$-variable L-function of Langlands is sketched in terms of monomial resolutions of admissible representations of reductive locally $p$-adic Lie groups.
Let $pi$ be an irreducible cuspidal automorphic representation of a quasi-split unitary group ${rm U}_{mathfrak n}$ defined over a number field $F$. Under the assumption that $pi$ has a generic global Arthur parameter, we establish the non-vanishing of the central value of $L$-functions, $L(frac{1}{2},pitimeschi)$, with a certain automorphic character $chi$ of ${rm U}_1$, for the case of ${mathfrak n}=2,3,4$, and for the general ${mathfrak n}geq 5$ by assuming a conjecture on certain refined properties of global Arthur packets. In consequence, we obtain some simultaneous non-vanishing results for the central $L$-values by means of the theory of endoscopy.
In [Ar13], Arthur classifies the automorphic discrete spectrum of symplectic groups up to global Arthur packets, based on the theory of endoscopy. It is an interesting and basic question to ask: which global Arthur packets contain no cuspidal automorphic representations? The investigation on this question can be regarded as a further development of the topics originated from the classical theory of singular automorphic forms. The results obtained yield a better understanding of global Arthur packets and of the structure of local unramified components of the cuspidal spectrum, and hence are closely related to the generalized Ramanujan problem as posted by Sarnak in [Sar05].
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