Symmetric square $L$-functions on $\mathrm{GL}_3$

Abstract: We give an asymptotic formula with a power-saving error term for the twisted first moment of symmetric square $L$-functions on $\mathrm{GL}_3$ in the spectral aspect. We apply this to obtain non-vanishing results and lower bounds of the expected order of magnitude for even moments, supporting the random matrix model for a unitary ensemble of $L$-functions. The main ingredients are the $\mathrm{GL}_3$ Kuznetsov formula, an asymmetric approximate functional equation, and strong bounds for the integral transforms appearing in the Kuznetsov formula.

• The paper establishes a precise asymptotic formula for the twisted spectral moment of symmetric square L-functions on GL3 with a power-saving error term.
• It employs the GL3 Kuznetsov formula, an asymmetric approximate functional equation, and two-dimensional stationary phase methods to control oscillatory integrals.
• The results provide robust quantitative non-vanishing and moment lower bounds that align with predictions from random matrix theory.

Asymptotic Analysis of Symmetric Square $L$-functions on $\mathrm{GL}_3$

Introduction and Context

The paper "Symmetric square $L$-functions on $\mathrm{GL}_3$" (2606.19959) performs a rigorous asymptotic analysis of the twisted first spectral moment of symmetric square $L$-functions arising from level-one cuspidal automorphic representations of $PGL_3$. The focus is on the spectral aspect, i.e., averaging over families parametrized by Langlands parameters $\mu_\pi$ in a ball of radius $T^\delta$ about a generic position $\tilde{\mu}$ in the spectrum. The symmetric square $L$-function, of degree six, is not known to be automorphic in full generality; however, its analytic properties (entireness except at $\mathrm{GL}_3$0 and functional equation) are derived via its relation to Rankin–Selberg $\mathrm{GL}_3$1-functions and recent work on the adjoint lift from $\mathrm{GL}_3$2 to $\mathrm{GL}_3$3 [Ga25].

The main result is a precise asymptotic formula for the twisted moment with a power-saving error term. This formula facilitates quantitative non-vanishing results and lower bounds for even moments, corroborating the predictions of random matrix theory for unitary ensembles of $\mathrm{GL}_3$4-functions.

Methodological Framework

The analytic approach deploys the $\mathrm{GL}_3$5 Kuznetsov formula in the spectral aspect (using the Li/Buttcane presentation), an asymmetric approximate functional equation tailored to circumvent challenges with archimedean oscillations, and stationary phase methods for controlling the geometric side. The test function $\mathrm{GL}_3$6 is constructed as a holomorphic approximation to the indicator of the family ball, with zeros placed to cancel archimedean poles arising in the functional equation.

Spectral parameters are normalized by $\mathrm{GL}_3$7 with generic positioning ensuring avoidance of Weyl chamber walls and self-dual forms (which would require separate argumentation due to altered analytic conductor). Hecke–Satake theory is used to parametrize the Fourier-Whittaker coefficients, which are multiplicative and bounded by the best-known bounds towards Ramanujan ($\mathrm{GL}_3$8).

The approximate functional equation is asymmetric, intentionally shortening the range of the second sum to minimize oscillatory interference in the critical regime. The main term in the moment formula appears only in the diagonal Weyl element, with the long Weyl element and associated kernel demanding a delicate analysis of oscillatory integrals—specifically a two-dimensional stationary phase argument. The integral transforms in the Kuznetsov formula are tightly bounded using recent advances in their Mellin–Barnes representations and asymptotics for associated Bessel functions.

Main Results

Asymptotic Moment Formula

The principal theorem establishes, for $\mathrm{GL}_3$9 in generic position and $L$0 as defined above: $L$1 where $L$2 is the spectral volume, and $L$3 is the normalizing factor closely related to the adjoint $L$4-function.

Non-vanishing and Moment Lower Bounds

A robust quantitative non-vanishing result is obtained: for $L$5 and $L$6 generic, at least $L$7 forms $L$8 exist in a $L$9 neighborhood of $\mathrm{GL}_3$0 with $\mathrm{GL}_3$1. For the moment problem, the lower bound for the $\mathrm{GL}_3$2-th moment matches the random matrix theory exponent: $\mathrm{GL}_3$3 demonstrating consistency with the unitary ensemble.

Technical Highlights

Diagonal versus Off-diagonal

The diagonal elements yield the main term as expected; however, intricate control over the off-diagonal (especially the contribution of the long Weyl element) is achieved by truncating to critical ranges via decay properties of the $\mathrm{GL}_3$4 integral transforms and further suppressing negligible contributions through stationary phase, Poisson summation, and nearly square-root cancellation for the underlying exponential sums (Gauss sums).

Treatment of Archimedean and Arithmetic Difficulties

The spectral aspect places primary analytic difficulties in the management of archimedean oscillatory integrals and the nontrivial action of the Weyl group. Arithmetic complications from finite ramified places are absent at level one, though the analytic complexity of higher-rank Kloosterman sums remains. The test functions, their zeros, and the normalization factors (relating back to $\mathrm{GL}_3$5) are chosen to ensure absolute convergence and rapid decay away from the spectral ball.

Approximate Functional Equation and Stationary Phase

The formulation of the asymmetric approximate functional equation, with a polynomial to cancel archimedean poles, sidesteps issues arising from additional oscillatory terms and shorter ranges in the second summand. The two-dimensional stationary phase argument for handling $\mathrm{GL}_3$6 achieves the necessary power-saving, enabling error terms of the form $\mathrm{GL}_3$7.

Implications and Future Directions

The results provide substantial progress in the analytic theory of spectral moments for higher-rank automorphic $\mathrm{GL}_3$8-functions, specifically $\mathrm{GL}_3$9, supplementing the body of work on level and prime-aspect families [BC25]. The non-vanishing and moment bounds support the paradigm that random matrix theory models furnish accurate predictions even for intricate spectral families in higher rank, where the spectral measure is less tractable than in $L$0.

Practically, these advances enable quantitative control of central values and moments in $L$1, which is essential for arithmetic applications related to automorphic forms and the Langlands program, such as equidistribution, subconvexity, and potential advances in the understanding of automorphy and functoriality for symmetric square lifts.

Theoretically, several avenues remain open: extending the analysis to more general families (including higher level, or non-$L$2 base fields), refining the error terms (e.g., optimizing the $L$3 exponent), and generalizing to higher symmetric powers or other $L$4-functions attached to $L$5 with $L$6. Interaction with the random matrix models for non-unitary ensembles and further exploration of the implications for zero-density and subconvexity are also promising.

Conclusion

The study presents a detailed asymptotic formula with power-saving error term for the twisted spectral moment of symmetric square $L$7-functions on $L$8, leveraging advanced analytic tools such as the Kuznetsov formula, specialized test functions, and stationary phase methods. These results facilitate quantitative non-vanishing and establish moment lower bounds in line with random matrix theory, representing substantial analytic progress in higher rank automorphic $L$9-function theory. The technical innovations in handling archimedean integrals and spectral localization are likely to inform further developments in analytic number theory and automorphic forms.