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Twisted Moments of Rankin-Selberg $L$ -functions in the Prime-Power Level Aspect

This paper computes the twisted first and second moments of the shifted central values of Rankin-Selberg $L$ -functions $L(\frac{1}{2}+\omega, f\otimes g)$ as the primitive form $f$ varies over prime power levels $p^\nu$ with $\nu \geq 3$ , while $g$ remains a fixed primitive form of level coprime to $p$ .

Original authors: Fatma Çiçek, Alia Hamieh

Published 2026-06-08

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CC BY 4.0

Original authors: Fatma Çiçek, Alia Hamieh

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

Imagine you are a detective trying to understand the hidden "DNA" of a vast collection of mysterious musical instruments. In the world of mathematics, these instruments are called modular forms, and they produce complex sounds known as L-functions. These sounds aren't just noise; they contain deep secrets about numbers, primes, and the structure of the universe.

This paper is like a detailed acoustic analysis of a specific, very crowded room full of these instruments. Here is the story of what the authors, Fatma Çiçek and Alia Hamieh, discovered, explained in simple terms.

The Setting: A Room of Instruments

Usually, mathematicians study these instruments in rooms where the level of "noise" (called the level in math terms) is simple, like a prime number (2, 3, 5, 7). But this paper looks at a much more chaotic room where the noise level is a prime power (like $p^3$ , $p^4$ , etc.).

Think of a prime power level as a room where the instruments are not just slightly out of tune; they are heavily "ramified" or tangled. This creates a much more complex acoustic environment. In this room, you have:

New Instruments: Fresh, unique sounds (called newforms).
Old Instruments: Sounds that are echoes of instruments from smaller, simpler rooms (called oldforms).

The authors had to figure out how to separate the new sounds from the old echoes, which is much harder in this "prime power" room than in a simple one.

The Experiment: Twisting the Sound

The authors wanted to measure the moments of these sounds. In everyday language, a "moment" is like taking a snapshot of the volume of the music at a specific pitch.

The First Moment: They asked, "If we play all these instruments together, what is the average volume?"
The Second Moment: They asked, "If we play them together and measure the energy (volume squared), what do we get?"

To make the experiment more precise, they "twisted" the sound. Imagine taking a recording and playing it slightly faster or slower (a "shift"). This helps them see the structure of the music more clearly, revealing patterns that are hidden when the music is played at the standard speed.

The Challenge: The "Old" Echoes

The biggest hurdle in this paper was the oldforms. In simpler rooms, you can ignore the echoes. But in this prime-power room, the echoes are so loud and numerous that they mess up the calculation.

The authors had to develop a new, intricate way to filter out these echoes (using a tool called the Petersson trace formula).
They had to account for the fact that the "old" instruments were actually copies of instruments from lower levels, just scaled up. This required a lot of complex arithmetic to untangle.

The Results: The Main Discoveries

After doing the heavy lifting, the authors found two main things:

1. The Average Volume (First Moment)
They calculated exactly what the average volume of the music is when the instruments are twisted.

The Formula: They wrote down a precise mathematical recipe (a formula) that predicts this average.
The Surprise: The formula includes a term that grows with the "depth" of the room (the power of the prime). This tells us that the more tangled the room is, the louder the average sound becomes in a very specific, predictable way.
A Detective Tool: They showed that if you know the average volume of the music for every instrument in the room, you can uniquely identify the "fixed" instrument ( $g$ ) that was being played alongside them. It's like saying, "If I know the average reaction of a crowd to a specific singer, I can tell you exactly who that singer is."

2. The Energy of the Music (Second Moment)
They also calculated the "energy" (the square of the volume).

The Polynomial: They found that the energy doesn't just grow randomly; it follows a specific curve (a polynomial) based on the size of the room.
The Leading Term: The most important part of this curve (the "leading coefficient") depends on a special property of the fixed instrument called the symmetric square L-function. This is a deep connection between the energy of the music and the internal structure of the instrument.

Why This Matters (According to the Paper)

The authors don't claim this will cure diseases or build bridges. Instead, they frame this as a victory for analytic number theory.

Isolating the Variable: By studying this specific "prime power" room, they isolated the effect of deep "ramification" (tangling) at a single prime number. This is like studying how a single specific type of noise affects a whole orchestra, rather than just looking at the orchestra as a whole.
Refining the Tools: They proved that even in these messy, complex rooms, you can still get precise answers if you account for the "old echoes" correctly.
Future Steps: They mention that their results can be used to prove that these L-functions are not zero (nonvanishing) and to find tighter bounds on how loud they can get (subconvexity), which are major goals in the field.

The Bottom Line

This paper is a masterclass in untangling a very messy mathematical knot. The authors took a complex, noisy environment (prime-power levels with oldforms), applied a "twist" to the data, and successfully calculated the average and energy of the resulting sounds. They showed that even in the most complicated settings, the universe of numbers follows a strict, predictable, and beautiful pattern.

Technical Summary: Twisted Moments of Rankin-Selberg L-Functions in the Prime-Power Level Aspect

Problem Statement
This paper investigates the asymptotic behavior of the twisted first and second moments of shifted central values of Rankin-Selberg $L$ -functions, specifically $L(\frac{1}{2} + \omega, f \otimes g)$ . The study focuses on the family where $f$ varies over the set of primitive newforms $H^*_k(p^\nu)$ of even weight $k$ and prime power level $p^\nu$ (with $\nu \ge 3$ ), while $g$ is a fixed primitive form of level $D$ (where $D=1$ or $D$ is prime) coprime to $p$ . The shift $\omega$ is a bounded complex number.

The authors highlight a distinct feature of the prime-power level aspect compared to prime or square-free levels: the space of cusp forms $S_k(p^\nu)$ contains a significant contribution from oldforms. This structural richness complicates the Petersson trace formula and introduces additional arithmetic considerations in moment computations, necessitating a more intricate analytical approach.

Methodology
The core of the methodology relies on the approximate functional equation to express the central $L$ -values as rapidly decaying series involving Fourier coefficients. The authors then apply the Petersson trace formula (specifically the version for prime power levels established by Rouymi) to evaluate the harmonic sums over the family $H^*_k(p^\nu)$ .

Key technical steps include:

Decomposition: The trace formula separates the sums into diagonal and nondiagonal contributions. The presence of oldforms in the prime-power setting requires careful handling of the diagonal terms, which involve subtracting contributions from lower levels.
Nondiagonal Estimation: The nondiagonal terms are bounded using Weil's bound for Kloosterman sums, Voronoï summation formulas, and large sieve inequalities (Deshouillers-Iwaniec and Matomäki).
Off-diagonal vs. Off-off-diagonal: The nondiagonal sums are further split into "off-diagonal" and "off-off-diagonal" parts. The off-diagonal terms are evaluated explicitly using contour integration and residue calculus, while the off-off-diagonal terms are shown to be negligible error terms using bounds on shifted convolution sums of Fourier coefficients.
Cancellation: A crucial aspect of the proof involves demonstrating that certain main terms arising from the diagonal and off-diagonal contributions cancel each other out, leaving a specific polynomial structure in the shift parameters.

Key Contributions and Results

Theorem 1.1 (Twisted First Moment): The authors derive an asymptotic formula for the twisted first moment $\sum_{f} \lambda_f(\ell) L(\frac{1}{2} + \omega, f \otimes g)$ . The formula includes explicit main terms involving the shift $\omega$ , the level $p^\nu$ , and the fixed form $g$ , along with fully specified error terms depending on $p, \nu, \ell,$ and the weights.
- Corollary 1.2: By taking the limit $\omega \to 0$ , the authors obtain the asymptotic for the unshifted first moment. This result is used to prove a determination theorem: a primitive form $g$ is uniquely determined by the family of central values $\{L(\frac{1}{2}, f \otimes g)\}_{f \in H^*_k(p^\nu)}$ . This extends previous determination results to the prime-power level setting, where oldform contributions are non-trivial.
Theorem 1.3 (Twisted Second Moment): The paper presents an asymptotic formula for the twisted second moment $\sum_{f} \lambda_f(\ell) L(\frac{1}{2} + \omega_1, f \otimes g) L(\frac{1}{2} + \omega_2, f \otimes g)$ . The main term is expressed as a sum of four components involving the function $m(\omega_1, \omega_2)$ , which incorporates the Rankin-Selberg convolution $L(s, g \otimes g)$ and specific arithmetic factors related to the prime power level.
- Corollary 1.4: Setting $\omega_1 = \omega_2 = 0$ , the authors derive the asymptotic for the second moment at the central point. The main term is a cubic polynomial in $\nu \log p$ . The leading coefficient is explicitly computed in terms of the symmetric square $L$ -function $L(1, \text{sym}^2 g)$ and arithmetic factors involving the Euler totient function and the level $p$ .

Significance and Context
The paper positions itself within the broader context of analytic number theory, specifically the study of moments of automorphic $L$ -functions, which are central to problems regarding nonvanishing, subconvexity, and the distribution of zeros.

Comparison to Existing Literature: The authors note that while Duke, Akbary, Kowalski-Michel-VanderKam, and Balkanova have established results for moments of $L$ -functions at prime or square-free levels, the prime-power level case presents unique challenges due to the oldform structure. The results here are analogous to Balkanova's work on the fourth moment of $L(\frac{1}{2}, f)$ for prime power levels but extend the scope to Rankin-Selberg convolutions.
Methodological Rigor: The paper emphasizes that the derived asymptotic formulae include explicit, non-uniform error terms with fully specified dependence on all parameters ( $p, \nu, k, k', \ell, \omega$ ). This precision is noted as a distinctive feature compared to some prior works.
Applications: The authors state that the established second moment formula can be utilized to derive subconvexity and nonvanishing results for the family of values $\{L(\frac{1}{2} + \omega, f \otimes g)\}$ . The determination result (Corollary 1.2) serves as an application showing the uniqueness of the fixed form $g$ based on the moment data.

The work is described as "entirely analytic in nature," focusing on the technical intricacies required to isolate the effects of deep ramification at a fixed prime within the framework of Rankin-Selberg convolutions.