On the non-vanishing of the central value of certain $L$-functions: unitary groups

Jiang, Dihua; Zhang, Lei

doi:10.4171/jems/955

Cited by 8 publications

(3 citation statements)

References 14 publications

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“…In Section 7.3, we consider the situation that a cuspidal automorphic member π in Π φ (G n ) has the property that p m (π) = {p subr }, where p subr is the partition associated to the subregular nilpotent orbit, and prove in Proposition 7.3 that Conjecture 2.3 holds for this case. Further discussions on the Generic Summand Conjecture: its variants and applications can be found in our work ( [44] and [46]). In [45], we establish the local analogy of the Generic Summand Conjecture for orthogonal groups defined over p-adic local fields of characteristic zero.…”

Section: Discrete Spectrum and Fourier Coefficientsmentioning

confidence: 99%

Arthur parameters and cuspidal automorphic modules of classical groups

Jiang

Zhang

2020

Ann. of Math. (2)

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The endoscopic classification via the stable trace formula comparison provides certain character relations between irreducible cuspidal automorphic representations of classical groups and their global Arthur parameters, which are certain automorphic representations of general linear groups. It is a question of J. Arthur and W. Schmid that asks: How to construct concrete modules for irreducible cuspidal automorphic representations of classical groups in term of their global Arthur parameters? In this paper, we formulate a general construction of concrete modules, using Bessel periods, for cuspidal automorphic representations of classical groups with generic global Arthur parameters. Then we establish the theory for orthogonal and unitary groups, based on certain well expected conjectures. Among the consequences of the theory in this paper is that the global Gan-Gross-Prasad conjecture for those classical groups is proved in full generality in one direction and with a global assumption in the other direction.

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Section: Discrete Spectrum and Fourier Coefficientsmentioning

confidence: 99%

Arthur parameters and cuspidal automorphic modules of classical groups

Jiang

Zhang

2020

Ann. of Math. (2)

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“…2 Recently, Dihua Jiang and Lei Zhang [JZ20] have confirmed this conjecture when n 4. Of course, when n 2, it was already known before.…”

Section: If We Denote By λ *mentioning

confidence: 85%

Fourier–Jacobi cycles and arithmetic relative trace formula (with an appendix by Chao Li and Yihang Zhu)

Liu

Zhu

2021

Cambridge Journal of Mathematics

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In this article, we develop an arithmetic analogue of Fourier-Jacobi period integrals for a pair of unitary groups of equal rank. We construct the so-called Fourier-Jacobi cycles, which are algebraic cycles on the product of unitary Shimura varieties and abelian varieties. We propose the arithmetic Gan-Gross-Prasad conjecture for these cycles, which is related to central derivatives of certain Rankin-Selberg L-functions, and develop a relative trace formula approach toward this conjecture. As a necessary ingredient, we propose the conjecture of the corresponding arithmetic fundamental lemma, and confirm it for unitary groups of rank at most two and for the minuscule case. YIFENG LIU B.2. Main theorem and consequences 58 B.3. Proof of Theorem B.4 60 Appendix C. Shimura varieties for hermitian spaces 64 C.1. Case of isometry 64 C.2. Case of similitude 66 C.3. Their connection 68 C.4. Integral models and uniformization 70 Appendix D. Cohomology of unitary Shimura curves 75 D.1. Oscillator representations of local unitary groups 75 D.2. Setup for cohomology of Shimura varieties 77 D.3. Statements for cohomology of unitary Shimura curves 78 D.4. Proof of Theorem D.6 81 References 85

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“…are used to proved one direction of the global Gan-Gross-Prasad conjecture unitary groups and special orthogonal groups in a uniform way, and prove the other direction of the global Gan-Gross-Prasad conjecture with a global assumption that can only be verified in some special cases, currently. (4) In [19], the idea and method of the twisted automorphic descents was used to prove the non-vanishing of certain tensor product L-functions for classical groups. (5) In [13], the twisted automorphic descents were used to test the Branching Problem and the Reciprocal Branching Problem for cuspidal automorphic representations of classical groups.…”

Section: Introductionmentioning

confidence: 99%