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The Ranking-Selberg integral on ${\bf GSp(2)}$ for square free levels

This paper explicitly computes the Rankin-Selberg integral over adeles for vector-valued Siegel cusp forms of square-free levels by deriving exact evaluations of local zeta-integrals for specific test functions in Bessel models.

Original authors: Seiji Kuga, Masao Tsuzuki

Published 2026-04-29

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CC0 1.0

Original authors: Seiji Kuga, Masao Tsuzuki

Original paper dedicated to the public domain under CC0 1.0 (http://creativecommons.org/publicdomain/zero/1.0/). ✨ 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 trying to understand the hidden rhythm of a massive, complex musical orchestra. In the world of mathematics, this orchestra is made up of "Siegel cusp forms"—highly sophisticated, multi-dimensional wave patterns that exist in a space called the "symplectic similitude group" (a fancy name for a specific type of geometric symmetry).

The paper by Seiji Kuga and Masao Tsuzuki is like a detailed score that finally explains exactly how to measure the volume and harmony of these waves when they are played under specific, "square-free" conditions (think of this as playing the music without any muddy, overlapping notes).

Here is a breakdown of their journey using everyday analogies:

1. The Goal: Tuning the Orchestra

The authors want to calculate a specific "average" of how these musical waves interact with a special kind of background noise called an "Eisenstein series."

The Analogy: Imagine you have a choir (the cusp forms) singing in a large hall. You want to know how loud they are on average when they sing a specific note, but you also want to know how their voice blends with a specific echo (the Eisenstein series) bouncing off the walls.
The Challenge: The choir is huge, and the notes are complex. The authors needed a precise formula to calculate this "blending" (called a Rankin-Selberg integral) without getting lost in the math.

2. The Tool: The "Bessel Period" (The Special Microphone)

To measure the choir, they use a tool called a "Bessel period."

The Analogy: Think of a standard microphone that picks up all sound. A Bessel period is like a highly specialized, directional microphone that only picks up a very specific "vibration" or pattern within the sound. It filters out the noise to hear the pure essence of the wave.
The Discovery: The authors figured out exactly how to calibrate this microphone for their specific type of choir (vector-valued forms) and proved that if you use this microphone, you can break the complex global sound down into a product of tiny, local sounds (like listening to each instrument individually and multiplying their volumes).

3. The Method: The "Local-Global" Recipe

The paper uses a strategy called the "local-global method."

The Analogy: Imagine you want to know the total weight of a giant, mysterious cake. Instead of weighing the whole thing at once (which is hard), you cut it into small, manageable slices (local pieces). You weigh each slice perfectly, then multiply them all together to get the total weight.
The Math: The authors did this by looking at the "local" behavior of their waves at every prime number (like 2, 3, 5, etc.) and at infinity. They calculated the "local zeta-integrals" (the weight of each slice) for different types of musical notes (representations).
- Some slices were easy (unramified).
- Some were tricky because the notes were "ramified" (distorted or complex). The authors spent a lot of time in the paper (Sections 6 and 7) figuring out the exact weight of these tricky slices.

4. The Result: A Clear Formula

After doing all this heavy lifting, they produced a master formula.

The Analogy: They finally wrote down the exact recipe for the cake's weight. This recipe doesn't just give a number; it tells you that the total volume depends on three main ingredients:
1. The Spinor L-function: A mathematical "fingerprint" of the wave's complexity.
2. The Fourier Coefficients: The specific "notes" or frequencies that make up the wave.
3. Local Periods: The specific volume of each local slice.

5. The "Old" vs. "New" Distinction

One of the most important findings is about "old forms" versus "new forms."

The Analogy: Imagine the choir has "veteran singers" (old forms) who have been singing the same song for a long time, and "new singers" (new forms) who are just starting.
The Finding: As the "level" of the choir (the complexity of the rules they follow) gets bigger and bigger (going to infinity), the contribution of the "veteran singers" and some specific types of "guest singers" (Yoshida and Saito-Kurokawa lifts) fades away to zero.
Why it matters: This means that if you look at a very large, complex choir, the average sound is entirely determined by the "new" singers. The old ones become statistically invisible. This is a crucial step for the authors' future work, where they plan to use this result to solve a bigger geometric puzzle (a "relative trace formula").

Summary

In short, Kuga and Tsuzuki built a precise mathematical machine to measure the "volume" of complex, multi-dimensional waves. They figured out how to calibrate their measuring tools for every possible scenario, proved that the "old" waves disappear in the limit of large complexity, and provided a clear, explicit formula that connects the local details of these waves to their global behavior. This work serves as a foundational key for unlocking deeper mysteries in the arithmetic of these mathematical objects.

1. Problem Statement and Context

The paper addresses the explicit computation of a Rankin-Selberg type integral introduced by Piatetski-Shapiro for vector-valued Siegel cusp forms of square-free levels $\Gamma_0(N)$ on the symplectic similitude group $G = GSp(2)$.

Motivation: The Bessel periods on $GSp(2)$ are central to the arithmetic of Siegel modular forms, particularly in the context of Böcherer's conjecture (refined by Liu and solved by Furusawa-Morimoto). This conjecture relates Bessel periods to special values of $L$ -functions.
Specific Goal: The authors aim to compute the average of the inner product between an Eisenstein series $E(s, \Lambda, \mu)$ and Siegel cusp forms $\phi$ , weighted by their Bessel periods. Specifically, they investigate the asymptotic behavior of this average as the level $N$ (a square-free integer) tends to infinity, while the weight $l$ remains fixed.
Target Application: These results are foundational for a forthcoming paper [20] where the authors intend to analyze the geometric side of a relative trace formula. A critical requirement is to show that "old forms," "Yoshida lifts," and "Saito-Kurokawa lifts" contribute negligibly (vanish asymptotically) to this average, isolating the contribution of "general type" forms.

2. Methodology

The authors employ a local-global method, combining global automorphic theory with explicit local computations.

A. Global Setup

Rankin-Selberg Integral: They define the integral $\langle E(s, \Lambda, \mu), \phi \rangle$ over the adelic quotient of a subgroup $G^\#$ associated with an imaginary quadratic field $E$ .
Basic Identity: Using a lemma by Piatetski-Shapiro, they relate this global inner product to the Mellin transform of the global Bessel function $B_{T_\theta, \Lambda}(\phi)$ .
Decomposition: The global Bessel period is decomposed into an Euler product of local Bessel functions. This relies on the local multiplicity one theorem for Bessel models.
Newforms: They define a "newform" $\phi_\pi^0$ for each automorphic representation $\pi$ by fixing a system of local Bessel data $(\ell_p, \xi_p)$ , where $\ell_p$ is a local Bessel functional and $\xi_p$ is a local new vector, normalized such that $\ell_p(\xi_p)=1$ .

B. Local Computations (The Core Technical Work)

The paper provides explicit evaluations of local zeta-integrals for various types of irreducible admissible representations of $G(\mathbb{Q}_p)$ .

Representations: They focus on representations of Type I, IIb, IIIa, and VIb (classification by Schmidt/Atobe), which correspond to the local components of global forms with square-free level.
Test Functions: They utilize specific Schwartz-Bruhat functions and Bessel vectors (often new vectors or fixed vectors under congruence subgroups $K_0(p)$ ).
Cases Covered:
- Unramified cases: Standard computations.
- Ramified cases: Explicit calculations for representations with non-trivial conductor, including cases where the character $\Lambda$ or the level $N$ introduces ramification.
- Oldforms: Computations for representations that are not new at a prime $p$ (Type I and IIb with trivial central character).
- Newforms: Computations for Type IIIa and VIb newforms.
Gross-Prasad Periods: In Section 7, they compute a specific local period for ramified characters, extending previous results by Dickson et al. to the ramified setting.

3. Key Contributions and Results

A. Explicit Formula for the Average (Theorem 5.2 & 5.3)

The authors derive a completely explicit formula for the spectral average $I(s)(\lambda, N, f_S)$ . The result expresses the average in terms of:

Spinor L-functions: $L(s, \pi, \mu)$ twisted by a character $\mu$ .
Fourier Coefficients: A quantity $R(\phi_\pi^0, E, \Lambda)$ , which is an average of Fourier coefficients of the newform over the ray class group of $E$ .
Local Periods: Explicit factors $t(\pi_p, \mu_p)$ arising from the local zeta-integrals.

They refine this formula by separating the sum over automorphic representations into disjoint subsets:

General Type ( $G$ ): Forms not arising from lifts.
Yoshida Lifts ( $Y$ ): Lifts from pairs of $GL(2)$ forms.
Saito-Kurokawa Lifts ($SK$): Lifts from $PGL(2)$ forms.
Oldforms: Forms induced from lower levels.

B. Asymptotic Vanishing Results (Theorem 5.4)

A major contribution is the proof that as $N \to \infty$ (with $N$ prime and inert in $E$ ):

The contribution of old forms decays as $O(N^{-8})$ .
The contribution of Yoshida lifts decays as $O(N^{-3/2})$ .
The contribution of Saito-Kurokawa lifts vanishes (or decays as $O(N^{-3/2})$ depending on $\mu$ ).
Consequently, the average is dominated by the general type forms.

C. Functional Equations and Local Factors

They deduce the functional equation for the twisted spinor L-function $L(s, \pi, \mu)$ directly from the Rankin-Selberg integral, confirming that the local factors defined by Piatetski-Shapiro coincide with those in the literature (e.g., [38]).
They provide explicit formulas for the local $\epsilon$ -factors and local periods for all relevant types of representations, including the difficult ramified cases.

4. Significance

Resolution of Technical Obstacles: The paper fills a critical gap in the literature by providing explicit local computations for vector-valued forms and ramified characters. Previous works (e.g., [7]) largely handled scalar-valued forms or unramified cases.
Foundation for Trace Formulas: The results are essential for the authors' upcoming work on the relative trace formula. By proving that "bad" contributions (old forms, lifts) vanish asymptotically, they validate the use of this specific average to isolate the spectral contribution of general type forms, which is necessary for proving subconvexity bounds or non-vanishing results for L-values.
Verification of Conjectures: The explicit formulas serve as a rigorous check for the refined Gan-Gross-Prasad conjecture in the context of Siegel modular forms, extending the scope to square-free levels and vector-valued forms.
Computational Toolkit: The detailed tables and formulas for local zeta-integrals (Sections 6 and 7) provide a valuable reference for future researchers working on automorphic forms on $GSp(2)$ and related groups.

In summary, Kuga and Tsuzuki provide a rigorous, explicit, and asymptotic analysis of Rankin-Selberg integrals for Siegel cusp forms, bridging the gap between abstract automorphic representation theory and concrete arithmetic applications involving L-functions and Bessel periods.