Non-Vanishing of Cubic Twists of $GL_n(\mathbb{Q})$ $L$-functions

Abstract: Let $π$ be an irreducible, cuspidal automorphic representation of $GL_n(\mathbb{A}_\mathbb{Q})$ ($n\geq 3$), which is tempered only for $n=3$. Let $s$ be a complex number such that $\Re(s)\notin \left[1/n, 1-1/n\right]$ if $n\neq 4$; $\Re(s)\notin\left[1/5, 4/5\right]$ if $n=4$, then we show that there are infinitely many primitive cubic Dirichlet characters $χ$ such that $L(s,π\times χ)\neq 0$. Similar results were previously known only for primitive Dirichlet characters without any restriction on the order and quadratic Dirichlet characters.

• The paper establishes non-vanishing for L(s, π×χ) by proving that infinitely many primitive cubic Dirichlet characters yield non-zero L-values for GLₙ automorphic forms when s is outside critical intervals.
• It uses advanced methods such as averaging the first moment, cubic large sieve inequalities, and careful optimization of error terms to bypass traditional limitations from hyper-Kloosterman sums.
• The result extends known quadratic twist cases to cubic twists for higher-rank groups, offering new insights into automorphic forms, arithmetic invariants, and the Langlands program.

Non-Vanishing of Cubic Twists of $GL_n(\mathbb{Q})$ $L$-functions

Introduction and Motivation

The non-vanishing of automorphic $L$-functions, especially when twisted by Dirichlet characters, remains a pivotal issue within analytic number theory. The properties of $L$-values are deeply connected to arithmetic invariants and automorphic forms. Evidencing non-vanishing in thin character families, such as primitive cubic Dirichlet characters, expands our understanding of both the analytic behavior and algebraic significance of these $L$-functions. Previous results addressed quadratic twists for $GL_n$ and cubic twists for $n=1$ with extensions to highly generic Dirichlet characters. However, the landscape for higher rank cases $(n\geq 3)$, and particularly for cubic twists, was previously unexplored due to technical barriers related to the thinness and arithmetic subtleties of these subfamilies.

Main Results

The paper establishes, for all $n\geq 3$, the existence of infinitely many primitive cubic Dirichlet characters $\chi$ such that the twisted $L$0-function $L$1 does not vanish at $L$2 outside a specific real interval for $L$3. More precisely, for $L$4 an irreducible cuspidal automorphic representation of $L$5, tempered for $L$6, and $L$7 with

• $L$8 if $L$9
• $L$0 if $L$1,

there exist infinitely many $L$2 of exact order 3 (primitive cubic Dirichlet characters) with $L$3. The restriction to tempered $L$4 for $L$5 is essential due to the analytic estimates required in the proof.

This result extends the framework of non-vanishing from quadratic to cubic twists for higher-rank groups. The analogous statement for quadratic characters was previously obtained for all $L$6 [see, e.g., CFH05], but cubic twists for $L$7, $L$8, were previously not covered outside the context of ground fields containing the relevant roots of unity (which is not the case for $L$9).

Technical Framework and Methods

The core of the proof combines advanced analytic methods:

• Averaging the first moment of the twisted $L$0-values over thin families of cubic characters, parameterized using cubic residue symbols on $L$1 (where $L$2 is a primitive third root of unity).
• Cubic large sieve inequalities: Central to the analysis, large sieve bounds for cubic characters generalize the classical quadratic sieve and are less sharp, necessitating new factorizations in the moduli of the twist.
• The authors choose factorizable moduli (for $L$3), allowing the sieve to be applied in a more effective way than in previous works, and double application of the sieve offers essential additional savings in error estimates.
• Approximate functional equations and precise control of main and error terms via complex analytic methods, drawing on the properties of Hecke $L$4-functions over $L$5 and their second moment bounds.

A prominent aspect of the approach is that it avoids heavy algebraic machinery such as bounds for hyper-Kloosterman sums. The analysis is constrained to the range of $L$6 away from the critical strip (for $L$7), echoing earlier limitations in non-vanishing results in general moduli. However, for $L$8, the factorizable moduli and application of the cubic large sieve provide an improvement in the effective range for $L$9.

The authors do not require the distribution of the sign of cubic Gauss sums, which were a technical obstruction in prior treatments for higher order twists.

Notable Numerical Strengths and Claims

• For $L$0 and $L$1, the main term in the first moment is shown to be $L$2 for a suitably chosen range of moduli parameters, guaranteeing non-vanishing for infinitely many cubic twists.
• For $L$3 and $L$4, the main term grows linearly in $L$5.
• The error terms, via careful optimization of the parameters and averaging, are always $L$6 for some explicit $L$7 depending on $L$8, $L$9, and the technical parameters defining the family.
• All results are unconditional (unlike some prior work where, e.g., the Lindelöf hypothesis was assumed for quartic twists), and the cubic character family is not restricted by the properties of the base field as in the multiple Dirichlet series methods.

Comparison with Prior Work

The work draws a sharp distinction with previous methods (e.g., the multiple Dirichlet series approach in [CFH05]) which required the inclusion of all $GL_n$0-th roots of unity in the ground field, a condition not satisfied for cubic characters over $GL_n$1. For instance, Chinta, Friedberg, and Hoffstein demonstrated non-vanishing for cubic Hecke characters only when the field contains all cube roots of unity, thus not covering the present situation.

Advances in this paper also subsume and extend the cases treated by Barthel and Ramakrishnan [BR94] and Luo [Luo05], particularly regarding the treatment of higher order (cubic) twists in the range beyond $GL_n$2.

Theoretical and Practical Implications

These non-vanishing results have implications for the arithmetic of automorphic forms and the Langlands program, notably regarding period integrals, special values, and the arithmetic of motives. Non-vanishing for thin families, such as cubic characters, is essential in the study of the analytic and algebraic multiplicity of automorphic forms and in understanding the distinct phenomena present in higher rank or higher order settings.

The methods suggest further possible progress:

• Extending non-vanishing to central values and/or even thinner families,
• Generalizing the double application of cubic large sieve estimates to other thin character families (e.g., quartic or higher order),
• Applying the techniques for cubic twists to related problems in analytic or algebraic number theory, particularly where large sieve estimates are lossy and have previously obstructed progress.

From a practical perspective, the result provides a guarantee for the non-triviality of automorphic $GL_n$3-functions for higher rank groups within these cubic twist families, a prerequisite for many further arithmetic applications.

Conclusion

The paper makes a substantial contribution to the theory of automorphic $GL_n$4-functions by establishing the non-vanishing of $GL_n$5 for infinitely many primitive cubic Dirichlet characters $GL_n$6 and automorphic representations $GL_n$7 of $GL_n$8, $GL_n$9, for $n=1$0 outside critical intervals. By utilizing analytic techniques adapted to the cubic character family and introducing novel averaging and sieve strategies, the authors resolve an outstanding case in the non-vanishing problem beyond quadratic twists. The methods not only bypass some obstructions inherent in prior algebraic techniques but open the path for further developments concerning non-vanishing over thin character families and potentially more general contexts in the theory of automorphic $n=1$1-functions.