arXiv subjects
Publications and source records attributed to Jian-Shu Li.
Using the modular symbol approach, we prove the rationality and period relations for certain critical values of Rankin-Selberg L-functions for $\mathrm{GL}(n)\times\mathrm{GL}(n)$ over any number field that contains a CM field.
Let $\mathsf k$ be a local field. Let $I_ν$ and $I_{ν'}$ be smooth principal series representations of $\mathrm{GL}_n(\mathsf k)$ and $\mathrm{GL}_{n-1}(\mathsf k)$ respectively. The Rankin-Selberg integrals yield a continuous bilinear map $I_ν\times I_{ν'}\rightarrow \mathbb C$ with a certain invariance property. We study integrals over a certain open orbit that also yield a continuous bilinear map $I_ν\times I_{ν'}\rightarrow \mathbb C$ with the same invariance property, and show that these integrals equal the Rankin-Selberg integrals up to an explicit constant. Similar results are also obtained for Rankin-Selberg integrals for $\mathrm{GL}_n(\mathsf k)\times \mathrm{GL}_n(\mathsf k)$.
We formulate and prove the archimedean period relations for Rankin-Selberg convolutions for $\mathrm{GL}(n)\times \mathrm{GL}(n-1)$. As a consequence, we prove the period relations for critical values of the Rankin-Selberg L-functions for $\mathrm{GL}(n)\times \mathrm{GL}(n-1)$ over arbitrary number fields.
We prove the vanishing of certain low degree cohomologies of some induced representations. As an application, we determine certain low degree cohomologies of congruence groups.
Over a non-archimedean local field of characteristic zero, we prove the multiplicity preservation for orthogonal-symplectic dual pair correspondences and unitary dual pair correspondences.
We introduce the notion of the automorphic dual of a matrix algebraic group defined over $Q$. This is the part of the unitary dual that corresponds to arithmetic spectrum. Basic functorial properties of this set are derived and used both to deduce arithmetic vanishing theorems of ``Ramanujan'' type as well as to give a new construction of automorphic forms.