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Uniform upper bounds for global coefficients a^M(S′_j,γ) as the set of places grows

Develop uniform upper bounds for the Arthur global coefficients a^M(S′_j,γ) when the set of places S′_j grows without bound (under Condition (A2)), enabling termwise upper estimates on individual geometric contributions I_{M,γ}(h_j).

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Background

In analyzing the geometric side under Condition (A2), the authors aim to bound negligible terms. When the auxiliary set of places S′j increases unboundedly, they indicate that they lack upper bounds for the global coefficients aM(S′_j,γ), which prevents termwise estimates of I{G,γ}(h_j).

Overcoming this bottleneck by obtaining uniform bounds for aM(S′_j,γ) would allow finer control of geometric terms and potentially broader applicability of their asymptotic analysis.

References

If $\lim_{j\to\infty} |S_j| = \infty$, then we do not know any upper bounds of $aM(S'_j,\gamma)$, and so we avoid evaluating upper bounds of individual terms $I_{G,\gamma}(h_j).

— Asymptotic behavior for twisted traces of self-dual and conjugate self-dual representations of $\mathrm{GL}_n$ (2402.11945 - Takanashi et al., 19 Feb 2024) in Section “Under Condition (A2)”