[Paper Review] Gamma factors for Asai representations of ${ m GL}_2$

Shih-Yu Chen, Cheng Yao+1 more|arXiv (Cornell University)|Oct 30, 2018

Advanced Algebra and Geometry20 references4 citations

TL;DR

This paper establishes explicit formulas relating gamma factors for Asai representations of GL₂ over a quadratic extension E/F to both local zeta integrals (Rankin-Selberg and Piatetski-Shapiro–Rallis) and Weil-Deligne representations. It proves that the L-factors and epsilon factors from zeta integrals match those from the Langlands parameterization, with precise epsilon factor identities involving central characters, norm characters, and Langlands constants, completing the theory for n=1 and n=2 in both archimedean and non-archimedean cases.

ABSTRACT

Let $E$ be a quadratic semisimple extension of a local field $F$ of characteristic zero. We determine explicit relation between gamma factors for Asai representations of $R_{E/F}{ m GL}_{2/E}$ defined by the Weil-Deligne representations and local zeta integrals. When $E = F imes F$, the results were due to Henniart and Jacquet. We completed the theory in this article based on explicit calculation.

Motivation & Objective

• To complete the theory of gamma factors for Asai representations of GL₂ over a quadratic extension E/F by comparing zeta integral factors with those from Weil-Deligne representations.
• To resolve the case when E is a field (not F×F) and F is either archimedean or non-archimedean, extending prior results by Henniart, Jacquet, and others.
• To establish precise epsilon factor formulas involving central characters, norm characters, and the Langlands constant λE/F(ψ), crucial for p-adic L-function constructions.
• To prove a dichotomy criterion for trilinear forms on GL₂(F) using the epsilon factor at s=1/2.
• To provide explicit interpolation formulas for twisted triple product p-adic L-functions via Ichino’s formula and local period integrals.

Proposed method

• The authors use the local Langlands correspondence to associate Weil-Deligne parameters to irreducible admissible representations π of GL₂(E) and τ of GLₙ(F), forming the tensor representation (r∘ϕπ)⊗ϕτ.
• They define the Asai representation r as a tensor induction from GL₂(C)×GL₂(C) with Galois action permuting components based on whether σ|E is trivial.
• For n=1, they compare Rankin-Selberg zeta integrals LRS(s, Asπ⊗τ) and εRS(s, Asπ⊗τ,ψ,ξ) with the Galois L- and ε-factors LGal(s, Asπ⊗τ) and εGal(s, Asπ⊗τ,ψ).
• For n=2, they use Piatetski-Shapiro–Rallis zeta integrals LPSR(s, Asπ⊗τ) and εPSR(s, Asπ⊗τ,ψ,ξ), assuming π and τ are generic and τ is a subquotient of a principal series.
• They perform explicit calculations in both archimedean and non-archimedean cases, using the functional equation of zeta integrals and the uniqueness of GL₂-invariant functionals to identify the gamma factor.
• They derive the functional equation Z(M*f(s),W) = γRS(s,Asπ,ψ,ξ)Z(f(s),W) and use it to identify the gamma factor via known L-function normalization.

Experimental results

Research questions

• RQ1 How do the local L- and ε-factors defined via zeta integrals for Asai representations of GL₂(E) compare to those from the Weil-Deligne parameterization?
• RQ2 What is the precise relation between the epsilon factor from zeta integrals and the Galois epsilon factor for Asai L-functions in the case n=1?
• RQ3 What is the explicit formula for the twisted gamma factor in the n=2 case, and how does it relate to the Galois factor and central characters?
• RQ4 Under what conditions does a non-zero GL₂(F)-invariant trilinear form exist on π⊗τ, and how is this related to the epsilon factor at s=1/2?
• RQ5 How can these local factor identities be used to construct and interpolate twisted triple product p-adic L-functions?

Key findings

• For n=1 and generic π, the Rankin-Selberg L-factor LRS(s, Asπ⊗τ) equals the Galois L-factor LGal(s, Asπ⊗τ).
• The Rankin-Selberg epsilon factor satisfies εRS(s, Asπ⊗τ,ψ,ξ) = ωπ(ξ)ωτ(ξ²)|ξ²|_F^{s−1/2}λE/F(ψ)⁻¹εGal(s, Asπ⊗τ,ψ), explicitly relating the two factorizations.
• For n=2, the Piatetski-Shapiro–Rallis gamma factor satisfies γPSR(s, Asπ⊗τ,ψ,ξ) = ω(4ξ²)⁻¹|4ξ²|_F^{−2s+1}ωE/F(−1)γGal(s, Asπ⊗τ,ψ), with ω = ωπ|F×·ωτ.
• The dichotomy criterion holds: HomGL₂(F)(π⊗τ,ℂ) ≠ 0 if and only if ωE/F(−1)εGal(1/2, Asπ⊗τ) = 1.
• The results are essential for constructing twisted triple product p-adic L-functions, as they confirm non-vanishing and good p-adic behavior of local period integrals in Ichino’s formula.
• The functional equation for zeta integrals is proven via uniqueness of GL₂-invariant functionals and explicit computation of the intertwining operator M*.