Non-vanishing of certain integral representations

TL;DR

The paper establishes uniform test vectors in both Flicker and Bump–Friedberg integral frameworks, proving that the local zeta integrals $Z(s,W,\Phi)$ and $B(s_1,s_2,W,\Phi)$ do not vanish for all complex arguments when appropriate Whittaker and Schwartz data are chosen. It combines global Eisenstein/Flicker constructions with archimedean newform theory and local factorization to show non-vanishing across $\mathbb{C}$ (and $\mathbb{C}^2$) and to control the meromorphic behavior of corresponding partial $L$-functions. As corollaries, it derives precise pole-locusts for the partial Asai $L$-functions $L^S(s,\Pi,\mathrm{As})$ and for the partial Bump–Friedberg $L$-functions $L^T(s,\Pi,\mathrm{BF})$, depending on the central character $\omega_\Pi$ and the parity of $n$. The results provide uniform, globally valid non-vanishing test vectors and clarify the analytic structure of these $L$-functions within the global Langlands program.

Abstract

In this paper, we prove that there exist Whittaker and Schwartz functions such that the local Flicker integrals are non-vanishing for all complex values of $s$, and the local Bump-Friedberg integrals are non-vanishing for all complex pairs $(s_1,s_2)$. As a corollary, we determine the potential locations of poles for their corresponding partial $L$-functions.

Theorems & Definitions (16)

• Theorem 1.1
• Theorem 1.2
• Corollary 1.3
• Corollary 1.4
• Theorem 2.1
• Proposition 3.1
• Proposition 3.2
• Proposition 3.3
• Proposition 3.4
• Proposition 4.1