arXiv subjects
Publications and source records attributed to Zhengyu Mao.
In this paper, we give a complete list of strongly tempered hyperspherical Hamiltonian spaces. We show that the period integrals attached to the list contains many previously studied Rankin-Selberg integrals and period integrals, thus give a new conceptual understanding of these integrals. The list also proposes many new interesting period integrals to study.
We propose two families of relative trace formula comparisons in the study of relative Langlands duality conjectured by Ben-Zvi--Sakellaridis--Venkatesh. This allows us to incorporate numerous relative trace formula comparisons studied during the last four decades under the BZSV duality framework. For the proposed relative trace formula comparisons associated to some strongly tempered spherical varieties, we will prove the fundamental lemma and smooth transfer in the $p$-adic case. Moreover, inspired by the BZSV duality conjecture, we propose a conjecture regarding the degenerate Whittaker period, which generalizes Lapid-Mao's conjecture of the Whittaker period
We construct analogues of Rankin--Selberg integrals for Speh representations of the general linear group over a $p$-adic field. The integrals are in terms of the Shalika model and are expected to be the local counterparts of (suitably regularized) global integrals involving square-integrable automorphic forms and Eisenstein series on the general linear group over a global field. We relate the local integrals to the classical ones studied by Jacquet--Piatetski-Shapiro--Shalika. We also introduce a unitary structure for Speh representation on the Shalika model, as well as various other models including Zelevinsky's degenerate Whittaker model.
In previous papers we formulated an analogue of the Ichino--Ikeda conjectures for Whittaker--Fourier coefficients of automorphic forms on classical group and the metaplectic group. In the latter case we reduced the conjecture to a local identity. In this paper we will prove the local identity in the $p$-adic case, and hence the global conjecture under simplifying conditions at the archimedean places.
We study Whittaker--Fourier coefficients of automorphic forms on a quasi-split unitary group. We reduce the analogue of the Ichino--Ikeda conjectures to a conjectural local statement using the descent method of Ginzburg--Rallis--Soudry.
The formal degree conjecture relates the formal degree of an irreducible square-integrable representation of a reductive group over a local field to the special value of the adjoint $γ$-factor of its $L$-parameter. In this paper, we prove the formal degree conjecture for odd special orthogonal and metaplectic groups in the generic case, which combined with Arthur's work on the local Langlands correspondence implies the conjecture in full generality.
In a previous paper we formulated an analogue of the Ichino-Ikeda conjectures for Whittaker-Fourier coefficients of cusp forms on quasi-split groups, as well as the metaplectic group of arbitrary rank. In this paper we reduce the conjecture for the metaplectic group to a local conjectural identity. We motivate this conjecture by giving a heuristic argument for the case $\widetilde{SL}_2$. In a subsequent paper we will prove the local identity in the $p$-adic case.
We study the interplay between different models of the same irreducible representation of the $F$-points of a reductive group over a local field.
We formulate an analogue of the Ichino-Ikeda conjectures for the Whittaker-Fourier coefficients of automorphic forms on quasi-split reductive groups. This sharpens the conjectures of Sakellaridis-Venkatesh in the case at hand.
Let f be a newform of weight two, prime level p. If D is a fundamental discriminant, define the twisted L-function L(f,D,s) to be the L-function associated to the twist of f by the quadratic character of conductor D. In this paper we consider the question of computing the family of twisted central values {L(f,D,1) : |D| <= x} for some x, by using an explicit version of Waldspurger's formula relating the central values L(f,D,1) to the |D|-th Fourier coefficient of weigth 3/2 modular forms in Shimura correspondence with f.
We prove a generalization to the totally real field case of the Waldspurger's formula relating the Fourier coefficient of a half integral weight form and the central value of the L-function of an integral weight form. Our proof is based on a new interpretation of Waldspurger's formula in terms of equality between global distributions. As applications we generalize the Kohnen-Zagier formula for holomorphic forms and prove the equivalence of the Ramanujan conjecture for half integral weight forms and a case of the Lindelof hypothesis for integral weight forms. We also study the Kohnen space in the adelic setting.