Periods detecting Eisenstein series and sums of $L$-values I

TL;DR

The paper advances the relative Langlands program by proving that the automorphic period attached to a Hamiltonian variety with dual $M$-data in the form $M=T^*(reve{G}/reve{L})$ can, for certain cuspidal Eisenstein series, be expressed as a finite sum of $L$-values indexed by fixed points of the extended $L$-parameter on $reve{M}$. The authors develop a robust framework of canonical extensions for Rankin–Selberg periods across coranks 0, 1, and higher, using unfolding arguments, zeta integrals, and careful convergence analysis to relate periods to products and sums of $L$-functions. They provide precise formulations of the fixed-point sets and the associated non-linear $L$-functions $L(1,T_{\,\sigma}\check{X})$, and demonstrate that the regularized periods of Eisenstein series decompose into contributions from these fixed points, aligning with the conjectural predictions of BZSV and its number-field analogs. The results cover both $(n,n)$- and $(n,n+m)$-Eisenstein series, offering new instances where periods can be computed as finite $L$-value sums, thereby deepening our understanding of the relative Langlands correspondence and its arithmetic consequences. The techniques combine Rankin–Selberg theory, Eisenstein series, intertwining operators, and the machinery of Langlands spectral decomposition to yield explicit period–$L$-value identities with canonical regularization. The work has potential implications for trace formulas and the broader study of special values in the automorphic spectrum, contributing concrete evidence toward the BZSV conjectural framework in the number-field setting.

Abstract

We study the automorphic period associated to a $G$-Hamiltonian variety $M$ whose dual is $\check{M} = T^*(\check{G}/\check{L})$, where $\check{G}$ is a general linear group and $\check{L}$ is a Levi subgroup. For certain cuspidal Eisenstein series, we prove that their period is equal to a finite sum of special values of $L$-functions. This sum is indexed by the fixed points of the associated extended $L$-parameter on $\check{M}$, confirming a conjecture by Ben-Zvi-Sakellaridis-Venkatesh in this case.

Figures (1)

• Figure 1: The unipotent subgroups $\cU_r$, $\cU_r'$, $R_r$ and $C_r$

Theorems & Definitions (103)

• theorem 1: Theorem \ref{['thm:n_m_Eisenstein']}, rough form
• remark 1
• theorem 2
• proposition 1
• proposition 2: Proposition \ref{['prop:n_m_unfolding']}
• lemma 1
• proof
• lemma 2
• remark 2
• proof