[Paper Review] Bounds for global coefficients in the fine geometric expansion of Arthur's trace formula for GL(n)

Jasmin Matz|arXiv (Cornell University)|Aug 25, 2013

Advanced Algebra and Geometry19 references4 citations

TL;DR

This paper establishes explicit upper bounds for the absolute values of global coefficients $ a^M(\gamma,S) $ in the fine geometric expansion of Arthur's trace formula for $ \mathrm{GL}(n) $, using estimates on unipotent orbital integrals and zeta function derivatives. The key result bounds these coefficients by a power of the number field discriminant and a sum over local zeta derivatives, valid uniformly across all Levi subgroups and conjugacy classes.

ABSTRACT

We give upper bounds for the absolute value of the global coefficients appearing in the fine geometric expansion of Arthur's trace formula for GL(n).

Motivation & Objective

• To establish uniform upper bounds for the absolute values of global coefficients $ a^M(\gamma,S) $ in the fine geometric expansion of Arthur's trace formula for $ \mathrm{GL}(n) $.
• To address the lack of explicit formulas for these coefficients in general cases, especially for non-semisimple elements.
• To extend known bounds from unipotent and semisimple cases to arbitrary conjugacy classes via reduction to the unipotent case.
• To provide quantitative control over error terms in trace asymptotics for Hecke operators on $ \mathrm{GL}(n) $.

Proposed method

• Derive bounds for unipotent contributions using reduction theory over number fields and estimates on orbital integrals.
• Apply local zeta function estimates at finite places, particularly bounding $ \left| \frac{\zeta_{F,v}^{(s_v)}(1)}{\zeta_{F,v}(1)} \right| $ for $ s_v \geq 0 $.
• Use the structure of Levi subgroups and conjugacy classes in $ \mathrm{GL}_n $ to reduce general coefficients to unipotent ones via restriction of scalars.
• Employ the decomposition of regular elliptic elements into semisimple and unipotent parts, and relate coefficients across field extensions.
• Fix a canonical measure normalization to ensure uniformity in bounds across number fields.
• Apply the theory of weighted orbital integrals $ J_M^G(\gamma,f) $ and their dependence on $ a^M(\gamma,S) $.

Experimental results

Research questions

• RQ1 What is the growth rate of the global coefficients $ a^M(\gamma,S) $ in Arthur’s trace formula for $ \mathrm{GL}(n) $, as a function of the number field and the set of places $ S $?
• RQ2 How can one bound $ |a^M(\gamma,S)| $ uniformly across all Levi subgroups $ M \subseteq \mathrm{GL}_n $ and conjugacy classes $ \gamma \in M(F) $?
• RQ3 Can the coefficients for arbitrary $ \gamma \in \mathrm{GL}_n(F) $ be bounded by reducing to the unipotent case using algebraic number theory?
• RQ4 What role do derivatives of Dedekind zeta functions at $ s=1 $ play in controlling the size of these coefficients?
• RQ5 Is there a uniform bound in terms of the discriminant of the field and local zeta function derivatives?

Key findings

• For any number field $ F $ of degree $ d $ and discriminant $ D_F $, there exist constants $ \kappa(n,d) \geq 0 $ and $ C(n,d) \geq 0 $ such that $ |a^M(\mathcal{V},S)| \leq C D_F^\kappa \sum_{\sum s_v \leq \eta} \prod_{v \in S_{\text{fin}}} \left| \frac{\zeta_{F,v}^{(s_v)}(1)}{\zeta_{F,v}(1)} \right| $ for unipotent orbits $ \mathcal{V} \in \mathfrak{U}^M $.
• The bound depends on the discriminant $ D_F $, the number of places in $ S $, and the sum of derivatives of local zeta functions at $ s=1 $, with $ \eta = \dim \mathfrak{a}_0^M $.
• For general $ \gamma \in M(F) $, the coefficient $ a^M(\gamma,S) $ is bounded by a product of coefficients over field extensions, reducing the problem to the unipotent case via restriction of scalars.
• The bound for $ a^M(\gamma,S) $ is controlled by $ |\mathrm{discr}^{M_1}(\gamma_s)|_\infty^\kappa $, where $ \gamma_s $ is the semisimple part of $ \gamma $, and $ \kappa > 0 $ is arbitrary.
• The ratio $ \left| \frac{a^M(\gamma,S)}{a^{L^\gamma}(\gamma_s,S)} \right| $ is also bounded by the same discriminant and zeta derivative terms, showing consistency across Levi subgroups.
• The results provide a quantitative foundation for asymptotic trace formulas with uniform error terms in the context of $ \mathrm{GL}_n $, especially for Hecke operator traces.