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Fangyang Tian

Publications and source records attributed to Fangyang Tian.

6 recordsLinked to original sources
arXiv2026

Local Jacquet-Shalika integrals and modifying factors

Over any local field of characteristic zero, we develop a local theory of Jacquet-Shalika integrals that were originally introduced by Jacquet and Shalika to study exterior square local $L$-factors for $GL_m$. In particular, at least in the archimedean case, we establish a complete functional equation matching Artin local $\varepsilon$-factors. Central to the argument is a novel comparison strategy between Jacquet-Shalika integrals and open-orbit integrals associated to the Shalika subgroup.

math.NT↗
arXiv2024

Period Relations for Standard $L$-functions of Symplectic Type

This article is to understand the critical values of $L$-functions $L(s,Π\otimes χ)$ and to establish the relation of the relevant global periods at the critical places. Here $Π$ is an irreducible regular algebraic cuspidal automorphic representation of $\mathrm{GL}_{2n}(\mathbb A)$ of symplectic type and $χ$ is a finite order automorphic character of $\mathrm{GL}_1(\mathbb A)$, with $\mathbb A$ is the ring of adeles of a number field $\mathrm k$.

math.NT↗
arXiv2020

Archimedean Non-vanishing, Cohomological Test Vectors, and Standard $L$-functions of $\mathrm{GL}_{2n}$: Complex Case

The purpose of this paper is to study the local zeta integrals of Friedberg-Jacquet at complex place and to establish similar results to our recent work in the reall case joint with C. Cheng and D. Jiang. In this paper, we will (1) give a necessary and sufficient condition on an irreducible essentially tempered cohomological representation $π$ of $\mathrm{GL}_{2n}(\mathbb{C})$ with a non-zero Shalika model; (2) construct a new twisted linear period $Λ_{s,χ}$; (3) give a necessary and sufficient condition on the character $χ$ such that there exists a uniform cohomological test vector $v\in V_π$ (which we construct explicitly) for $Λ_{s,χ}$. As a consequence, we obtain the non-vanishing of local Friedberg-Jacquet integral at complex place. All of the above are essential preparations for attacking a global period relation problem.

math.RT↗
arXiv2019

Archimedean Non-vanishing, Cohomological Test Vectors, and Standard $L$-functions of $\mathrm{GL}_{2n}$: Real Case

The standard $L$-functions of $\mathrm{GL}_{2n}$ expressed in terms of the Friedberg-Jacquet global zeta integrals have better structure for arithmetic applications, due to the relation of the linear periods with the modular symbols. The most technical obstacles towards such arithmetic applications are (1) non-vanishing of modular symbols at infinity and (2) the existance or construction of uniform cohomological test vectors. Problem (1) is also called the non-vanishing hypothesis at infinity, which was proved by Binyong Sun, by establishing the existence of certain cohomological test vectors. In this paper, we explicitly construct an archimedean local integral that produces a new type of a twisted linear functional $Λ_{s,χ}$, which, when evaluated with our explicitly constructed cohomological vector, is equal to the local twisted standard $L$-function $L(s,π\otimesχ)$ as a meromorphic function of $s\in \mathbb{C}$. With the relations between linear models and Shalika models, we establish (1) with an explicitly constructed cohomological vector, and hence recovers a non-vanishing result of Binyong Sun via a completely different method. Our main result indicates a complete solution to (2), which will be presented in a paper of Dihua Jiang, Binyong Sun and Fangyang Tian with full details and with applications to the global period relations for the twisted standard $L$-functions at critical places.

math.RT↗
arXiv2018

On the Archimedean Local Gamma Factors for Adjoint Representation of $\mathrm{GL}_3$, Part I

Studying the analytic properties of the partial Langlands $L$-function via Rankin-Selberg method has been proved to be successful in various cases. Yet in few cases is the local theory studied at the archimedean places, which causes a tremendous gap to complete the analytic theory of the complete $L$-function. In this paper, we will establish the meromorphic continuation and the functional equation of the archimedean local integrals associated with D. Ginzburg's global integral for the adjoint representation of $\mathrm{GL}_3$. Via the local functional equation, the local gamma factor $Γ(s,π,\mathrm{Ad},ψ)$ can be defined. In a forthcoming paper, we will compute the local gamma factor $Γ(s,π,\mathrm{Ad},ψ)$ explicitly, which fills in some blanks in the archimedean local theory of Ginzburg's global integral.

math.NT↗