Global coefficient a^M(𝒮,γ) for elliptic semisimple elements with non‑trivial unipotent part

Determine the global coefficient a^M(𝒮,γ) appearing in Arthur’s invariant (twisted) trace formula for F‑elliptic semisimple elements γ in M with non‑trivial unipotent part, including its explicit dependence on the finite set of places 𝒮.

Background

In the fine geometric expansion of the twisted invariant trace formula, global coefficients a^M(𝒮,γ) weight the contributions of θ‑conjugacy classes. For semisimple γ that are F‑elliptic in M, these coefficients are known and 𝒮‑independent when the unipotent part is trivial; they vanish when γ_s is not F‑elliptic.

However, when γ_s is F‑elliptic and γ_u is non‑trivial, the authors note that the global coefficient is generally unknown and should depend on 𝒮. Clarifying or computing these coefficients would strengthen geometric control in twisted trace formula applications.