Beyond Endoscopy for $\mathrm{GL}_3(\mathbb{Q})$: Functional equation for the $L$-function of a cubic order

Abstract: The Beyond Endoscopy strategy, proposed by Langlands, aims to establish the principle of functoriality by analyzing the trace formula. Recently, Deng and Espinosa advanced this program for $\mathrm{GL}_3(\mathbb{Q})$ by isolating the contribution of the trivial representation from the elliptic regular part. Their work relies on a conjectural factorization formula for the $L$-function associated with a cubic order, which yields the functional equation for the completed $L$-function. In this paper, we provide an unconditional proof of this functional equation for every Gorenstein order in a cubic number field. As a consequence, their isolation of the trivial representation for $\mathrm{GL}_3(\mathbb{Q})$ becomes fully unconditional.

• The paper proves an unconditional functional equation for the L-function of Gorenstein cubic orders, eliminating previous conjectural dependencies.
• It employs explicit local factor analysis and combinatorial identities to derive a finite Euler product representation of these L-functions.
• The result enhances the Beyond Endoscopy approach for GL3(Q), paving the way for advanced applications in automorphic transfer and trace formula techniques.

Functional Equation for the $L$-Function of a Cubic Order and Its Role in Beyond Endoscopy for $\mathrm{GL}_3(\mathbb{Q})$

Introduction and Context

This paper addresses a fundamental analytic obstruction in the "Beyond Endoscopy" program for $\mathrm{GL}_3(\mathbb{Q})$, specifically the unconditional establishment of a functional equation for the $L$-function attached to a cubic order. Langlands' Beyond Endoscopy proposal seeks to isolate automorphic representations by analytic means within the trace formula, with the ultimate aim of achieving new cases of Langlands functoriality. The initial stage involves analyzing the elliptic regular part of the stable trace formula and canceling the contribution from the trivial representation. In the $\mathrm{GL}_2$ context, this approach was realized by Altuğ, who exploited special properties of certain order zeta functions and their functional equations.

Deng and Espinosa extended this to the $\mathrm{GL}_3$ case, but their arguments were contingent on a conjecture relating the Dedekind zeta function of a cubic order (in the sense of Yun) to a specific higher-rank $L$-function built from overorders and local factors. The principal achievement of the present work is the removal of this conjectural dependence: for all Gorenstein cubic orders, the author establishes the required analytic functional equation unconditionally. This result renders previous conditional statements in the $\mathrm{GL}_3$ case completely unconditional and sharpens the analytic analytic machinery available for the "Beyond Endoscopy" program in rank 3.

$L$-Functions for Orders and Their Functional Equations

The notion of an $L$-function for an order generalizes the classical Dedekind zeta function to situations where the underlying order is not necessarily maximal. The key definition, motivated by the required expansion of orbital sums in the trace formula, is: $\mathrm{GL}_3(\mathbb{Q})$0 where $\mathrm{GL}_3(\mathbb{Q})$1 is a $\mathrm{GL}_3(\mathbb{Q})$2-order in a cubic number field $\mathrm{GL}_3(\mathbb{Q})$3, the sum is over overorders, local terms $\mathrm{GL}_3(\mathbb{Q})$4 incorporate Gorenstein properties, and the numerators are constructed using local zeta functions of the order. The functional equation is formulated for the completed $\mathrm{GL}_3(\mathbb{Q})$5-function: $\mathrm{GL}_3(\mathbb{Q})$6 where the gamma factor incorporates the discriminant and archimedean invariants associated to $\mathrm{GL}_3(\mathbb{Q})$7 and $\mathrm{GL}_3(\mathbb{Q})$8.

Previously, the functional equation of $\mathrm{GL}_3(\mathbb{Q})$9 was only known under strong conjectures (notably, a relation between $\mathrm{GL}_3(\mathbb{Q})$0 and Yun's order zeta function $\mathrm{GL}_3(\mathbb{Q})$1). This paper demonstrates that such strong hypotheses are not needed if the focus is strictly on the functional equation itself.

Methodology: Reduction to Local Factor Analysis

A significant technical contribution of the paper is the reduction of the functional equation to the local functional equations of explicitly constructed local factors, $\mathrm{GL}_3(\mathbb{Q})$2. Exploiting a classification of overorders of $\mathrm{GL}_3(\mathbb{Q})$3 in terms of ideal class monoid data—developed by the author and collaborators in recent preprints for Bass and Gorenstein orders—the paper expresses $\mathrm{GL}_3(\mathbb{Q})$4 as a finite Euler product of local polynomials.

Crucially, the analysis provides:

• An explicit parametrization of overorders for each $\mathrm{GL}_3(\mathbb{Q})$5 in terms of splitting invariants of $\mathrm{GL}_3(\mathbb{Q})$6 at $\mathrm{GL}_3(\mathbb{Q})$7 and the structure of $\mathrm{GL}_3(\mathbb{Q})$8.
• Closed-form polynomial formulas for local factors in all ramification, inert, and split cases, extending and unifying those known for quadratic and maximal cases.

The core of the paper consists of case-by-case combinatorial and algebraic manipulations—guided by the order theory and local-to-global principles—that demonstrate the required functional symmetry for each $\mathrm{GL}_3(\mathbb{Q})$9. Sophisticated summation manipulations and telescoping identities translate the analytic functional equation into polynomial identities that are verified uniformly in each local configuration.

Main Theorem and Its Proof Architecture

Theorem (Unconditional Functional Equation):

For all Gorenstein cubic orders $L$0 in a cubic number field $L$1,

$L$2

This holds unconditionally, that is, without recourse to the conjectural relationship tying $L$3 to the order zeta function $L$4.

Key ingredients:

• Local explicitness and finiteness: Only finitely many primes affect the non-maximality and Gorenstein property, so the global verification reduces to a finite task.
• Combinatorial identification: For each local splitting type, the relevant sums are manipulated into forms symmetric under $L$5. This requires delicate index reparametrizations and the recognition of the symmetries inherent from Gorenstein order structure.
• Algebraic-analytical interface: The functional equation, although analytic in origin, rests here on exact polynomial identities arising from the ideal-theoretic description of overorders.

The overall proof is canonical in that it does not depend on the heavier machinery of local harmonic analysis or the full power of the trace formula, but rather hinges on a fine understanding of order structures and the corresponding combinatorics.

Implications for Beyond Endoscopy and Automorphic Transfer

The result has substantial implications for the analytic isolation of representations via trace formulae (the first step in Beyond Endoscopy for higher rank groups):

• It removes the last conditional component in the $L$6 analysis initiated by Deng and Espinosa, thereby allowing the unconditional application of Poisson summation and approximate functional equation methods in this context.
• The analytic approach to functoriality and the transfer of trace formulae can now be implemented in full generality for cubic orders, meaning that spectral decompositions previously unavailable are now accessible.
• The structure theory developed here may generalize to higher rank groups ($L$7 for $L$8), where order zeta functions and their functional equations are expected to play a central role.

The result also informs the explicit harmonic analysis of regular elliptic orbital integrals and the classification of associated automorphic representations, enabling progress in the analytic theory of $L$9-functions for orders and the endoscopic classification in degree three.

Perspectives and Further Directions

The removal of the conjectural obstruction for the $\mathrm{GL}_2$0 Beyond Endoscopy program sharpens the analytic tools for both trace formula applications and the study of special values of higher-rank order $\mathrm{GL}_2$1-functions. The methods of this paper are sufficiently explicit to allow effective computations and may be of use in point-counting problems and arithmetic statistics related to cubic rings and their moduli.

The extension to non-Gorenstein orders, general étale algebras, or higher-rank groups remains open, with the present work suggesting a pathway through combinatorial and order-theoretic analysis rather than appealing to spectral methods alone.

Conclusion

This paper establishes the unconditional functional equation for the $\mathrm{GL}_2$2-function associated with an arbitrary Gorenstein order in a cubic number field, thereby removing a standing analytic obstruction in Beyond Endoscopy for $\mathrm{GL}_2$3. The combination of fine algebraic parametrizations, local-global reductions, and combinatorial identities provides a robust framework for higher-rank trace formula analyses and enhances the explicit toolkit for automorphic $\mathrm{GL}_2$4-function theory (2607.03083).