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An asymptotic formula for a cubic moment of $GL_2$ $L$ -functions

The paper establishes an asymptotic formula with a power-saving error term for a cubic moment of self-dual $GL_2$ $L$ -functions, incorporating a small extra averaging over characters and confirming the main term conjectured by Conrey, Farmer, Keating, Rubinstein, and Snaith.

Original authors: Catinca Mujdei

Published 2026-09-14

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: Catinca Mujdei

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the vast landscape of modern mathematics, there exists a hidden architecture built from numbers that behave like waves. These are not the simple integers used for counting apples, but complex, infinite sequences known as L-functions. Mathematicians have long suspected that these functions hold the keys to understanding the deepest patterns of prime numbers, the building blocks of arithmetic. Just as a musician might study the volume of a note to understand the instrument that produced it, researchers study the "moments" of these L-functions—essentially, the average size of their values at a specific, critical point. By looking at these averages, particularly when they are cubed or raised to higher powers, mathematicians hope to reveal the underlying symmetry and structure of the number system itself. For decades, a specific set of predictions, known as the CFKRS recipe, has served as a map for what these averages should look like. However, navigating the terrain to prove these predictions has been fraught with obstacles, as the calculations often produce a confusing mix of expected results and unexpected, messy leftovers that refuse to cancel out.

This paper, authored by Catinca Mujdei, takes a significant step forward in clearing that path. The author focuses on a particularly difficult challenge: calculating the average of the cube of these L-functions for a specific family of objects called automorphic forms. These forms are highly symmetric mathematical structures that appear in the study of prime numbers. Previous attempts to solve this problem, while successful in providing upper limits, had failed to produce a precise formula that matched the long-standing predictions. The main hurdle was a collection of "degenerate" terms—mathematical leftovers that appeared in the calculations and seemed to threaten the accuracy of the final result. These terms were like noise in a signal, obscuring the true pattern the researchers were trying to hear.

Mujdei's breakthrough comes from a clever change in perspective. Instead of trying to analyze a single, rigid family of these mathematical forms, the author introduces a small amount of flexibility by averaging the results over a slightly larger group of related characters. Think of this as tuning a radio slightly to find a clearer signal; by averaging over this small neighborhood of possibilities, the chaotic noise of the degenerate terms is silenced. The paper demonstrates that when this averaging is applied, the messy leftovers cancel each other out perfectly, leaving behind a clean, precise formula. This formula matches exactly what the CFKRS recipe had predicted years ago, confirming that the theoretical map was correct all along.

The result is not just a confirmation of a guess, but a rigorous proof that includes a power-saving error term. In mathematical terms, this means the author has not only found the main answer but has also quantified exactly how close the approximation is, showing that the remaining error is small enough to be negligible for the purposes of the study. The work specifically addresses a scenario involving prime numbers raised to high powers, a case that had proven resistant to previous methods. By successfully isolating the main term and proving the cancellation of the difficult off-diagonal terms, the paper validates the heuristic approach for this complex cubic moment.

This achievement has implications beyond the immediate calculation. The ability to produce a precise asymptotic formula opens the door to new applications, such as proving that certain values of these L-functions are large or that they do not vanish at the center of their range. These properties are crucial for understanding the distribution of prime numbers and the behavior of the functions themselves. The paper achieves this by combining advanced techniques from harmonic analysis, including the use of trace formulas that relate sums over these forms to sums over other mathematical objects, with careful combinatorial arguments. The author shows that the cancellation of the problematic terms is not accidental but a structural feature of the problem, arising from the interplay between the diagonal and off-diagonal components of the calculation.

Ultimately, this work provides a definitive answer to a question that has lingered in the analytic theory of L-functions. It confirms that the main term conjectured by the community is indeed the dominant feature of the cubic moment, even in the presence of complex, ramified conditions. The paper does not merely suggest this is true; it proves it with a level of precision that allows for future extensions and refinements. By resolving the issue of the degenerate terms, the author has removed a significant sticking point that had hindered progress in this area, offering a clearer view of the intricate symmetries that govern these fundamental mathematical objects. The findings stand as a testament to the power of averaging techniques in taming the complexity of high-dimensional number theory, turning a previously intractable problem into a solved one with a clear, verified formula.

Technical Summary: An Asymptotic Formula for a Cubic Moment of $GL_2$ L-functions

Problem Statement
The paper addresses the problem of establishing a rigorous asymptotic formula for the cubic moment of self-dual $GL_2$ L-functions, specifically in the context of families studied by Petrow and Young. While previous works (e.g., Conrey–Iwaniec, Petrow, Petrow–Young) have provided upper bounds or formulas containing a main term and a "dual" error term, they have not fully confirmed the conjectured main term proposed by Conrey–Farmer–Keating–Rubinstein–Snaith (CFKRS). The primary obstacle in this area has been the identification and cancellation of "degenerate off-diagonal" terms that arise in the trace formula expansion. These terms often obscure the true main term or prevent the error term from achieving a power-saving bound relative to the conductor.

The specific object of study is the averaged cubic moment:
$\mathbb{E}_{\phi \pmod{p^b}} [M_{nh}(\chi\phi, \alpha)]$
where $M_{nh}$ is a cubic moment of non-holomorphic (Maass) forms twisted by a primitive Dirichlet character $\chi$ of conductor $p^a$ , averaged over a small family of characters $\phi$ modulo $p^b$ (with $b \le a/3$ ).

Methodology
The proof follows a classical analytic number theory approach, combining the approximate functional equation with the Kuznetsov trace formula, followed by a detailed analysis of diagonal and off-diagonal terms. The methodology proceeds through the following stages:

Approximate Functional Equation and Trace Formula: The cubic moment is expanded using the approximate functional equation, resulting in a triple sum over integers $n_1, n_2, n_3$ . The Kuznetsov trace formula is then applied to the spectral sum, decomposing the expression into a diagonal term (where $n_1 = n_2 n_3$ ) and off-diagonal terms (involving Kloosterman/Salié sums).
Averaging over Characters: A crucial step is the averaging over the character $\phi \pmod{p^b}$ . This averaging shortens the modulus of the dual sum in the off-diagonal terms from $p^a$ to $p^{a-b}$ . This reduction is critical for applying large-sieve estimates effectively.
Diagonal Analysis: The diagonal contribution is evaluated using Mellin-Barnes integral representations. The author decomposes this sum into the conjectured CFKRS main term and several "false" main terms (residues from shifting contours).
Off-Diagonal Analysis and Poisson Summation: The off-diagonal terms are processed using Poisson summation, transforming them into dual sums over integers $m_1, m_2, m_3$ . The author evaluates the resulting Gauss-type sums and generating functions.
Cancellation Mechanism: The core technical achievement lies in Section 8, where the author demonstrates that the "false" main terms arising from the diagonal analysis exactly cancel with specific off-diagonal terms (specifically those where $m_1 m_2 m_3 = 0$ ). This cancellation is achieved through intricate combinatorics involving Mellin inversion formulas for Bessel transforms and properties of Gamma functions.
Error Term Estimation: The remaining off-diagonal terms (where $m_1 m_2 m_3 \neq 0$ ) are bounded using stationary phase methods and large-sieve inequalities. The averaging over $\phi$ allows for a saving of a factor $p^b$ in these estimates.

Key Contributions and Results
The paper establishes the following main result (Theorem 1.1):

Asymptotic Formula: For an odd prime $p$ , integers $a, b$ with $b \le a/3$ , and spectral parameter $T = (p^a)^{o(1)}$ , the averaged cubic moment satisfies:
$\mathbb{E}_{\phi \pmod{p^b}} [M_{nh}(\chi\phi, \alpha)] = \sum_{\sigma \in \{\pm 1\}^3} \frac{1+\sigma}{2} MT_{nh}(\alpha, \sigma) + O_\epsilon(p^{-b + a\epsilon})$
Here, $MT_{nh}$ is the main term conjectured by CFKRS, and the error term exhibits a power-saving relative to the conductor $p^a$ , provided $b$ is sufficiently large relative to $a$ .
Confirmation of CFKRS Conjecture: The paper proves that the main term for this cubic moment matches the CFKRS recipe exactly. This is the first instance where the conjectured main term for the Petrow–Young cubic moment (even with the slight enlargement of the family via averaging) is fully recovered and shown to dominate the dual sum.
Cancellation of Degenerate Terms: The work provides a rigorous proof for the cancellation of degenerate off-diagonal terms in the non-holomorphic setting. The author notes that this cancellation arises from different terms in the calculation compared to the holomorphic case (Petrow [Pet15]), utilizing Mellin-Barnes representations and functional equations for Gamma functions.
Technical Hypothesis: The result relies on the condition $b \le a/3$ . The author explains that this hypothesis is required to ensure that the error terms arising from contour shifts (specifically those related to poles on the $-1/2$ vertical line) are dominated by the error term obtained from the large-sieve estimates on the dual moment.

Significance and Claims
The paper claims significance in several specific areas:

Resolution of a Sticking Point: The identification and cancellation of degenerate off-diagonal terms have been a persistent difficulty in the literature surrounding cubic moments. This work resolves this for the non-holomorphic case with a power-saving error term.
Comparison of Methods: The author suggests that their classical approach (using trace formulas and approximate functional equations) may offer advantages over the "period integral" approach used in generalizations to number fields (e.g., by Balkanova–Frolenkov–Wu and Nelson), which has struggled to extract the conjectured main term in similar settings.
Foundation for Applications: By establishing an asymptotic formula with an explicit main term (rather than just an upper bound), the result opens the door to applications such as:
- Using Soundararajan's resonance method to prove the existence of large central L-values within the family.
- Combining with mollification techniques to prove simultaneous non-vanishing results for central L-values.
- Adapting the framework to mixed cubic moments or moments involving $GL_2 \times GL_2$ Rankin–Selberg L-functions.

The paper remains modest regarding the scope of the result, noting that the condition $b \le a/3$ is a technical simplification and that the non-averaged case ( $b=0$ ) would require proving the cancellation of the dual sum directly, which remains out of reach of current methods. The work is presented as a confirmation of the CFKRS conjecture for a specific, slightly enlarged family of automorphic forms.

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