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Spectral Moment Formulae for $\hbox{GL}(3)\times \hbox{GL}(2)$ $\hbox{L}$ -functions II: The Eisenstein Case

This paper establishes an exact Motohashi-type identity linking the shifted cubic moment of $\hbox{GL}(2)$ $\hbox{L}$ -functions to the shifted fourth moment of $\hbox{GL}(1)$ $\hbox{L}$ -functions via period integrals of $\hbox{GL}(3)$ Eisenstein series, while providing an intrinsic automorphic explanation for the main terms that aligns with the CFKRS Moment Conjectures.

Original authors: Chung-Hang Kwan

Published 2026-03-17

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Chung-Hang Kwan

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 trying to solve a massive, cosmic puzzle. In the world of mathematics, specifically Number Theory, there are special numbers called L-functions. Think of these as the "DNA" or "fingerprint" of prime numbers. They contain hidden patterns that tell us how prime numbers are distributed, but they are incredibly complex and hard to read directly.

Mathematicians often try to understand these patterns by looking at "moments." If you imagine the L-functions as a musical chord, a moment is like measuring the volume, the harmony, or the resonance of that chord.

This paper, written by Chung-Hang Kwan, is the second part of a trilogy trying to solve a specific, very difficult puzzle: How do two different types of musical chords (L-functions) relate to each other?

Here is the breakdown of what the paper achieves, using simple analogies:

1. The Big Problem: Two Different Worlds

For a long time, mathematicians have studied two specific types of "chords":

The Cubic Moment: This involves a complex relationship between three different L-functions (like a chord with three notes).
The Fourth Moment: This involves the Riemann Zeta function (the most famous L-function) raised to the fourth power (like a chord with four notes).

Historically, these two worlds seemed completely separate. One was studied using one set of tools, and the other with a different set. But in the 1990s, a mathematician named Motohashi discovered a "secret tunnel" connecting them. He found a formula that said: If you measure the volume of the "Four-Note Chord," you can mathematically translate it to measure the volume of the "Three-Note Chord," and vice versa.

This is called Spectral Reciprocity. It's like discovering that if you know the weight of a cloud, you can instantly calculate the depth of the ocean beneath it.

2. The Old Way vs. The New Way

Previous attempts to prove this connection were like trying to cross a canyon by building a shaky bridge out of loose rocks. They used complicated, messy techniques (like "Kloosterman sums" or "Petersson formulas") that worked but were hard to understand and full of "error terms" (loose rocks that might fall).

Kwan's approach is different. Instead of building a shaky bridge, he builds a helicopter.

He uses a concept called Period Integrals. Imagine the L-functions as waves on a surface. A "period integral" is like taking a snapshot of the wave at a specific moment. Kwan realizes that if you look at these waves from a higher, "3-dimensional" perspective (using a group called GL(3)), the connection between the 3-note and 4-note chords becomes obvious and natural.

3. The "Eisenstein" Twist

This specific paper focuses on a special case called the "Eisenstein Case."

Think of the "Cusp Forms" (the standard L-functions) as soloists in an orchestra. They are unique and distinct.
The "Eisenstein Series" are like the background choir. They are built from simpler parts and represent a continuous, flowing background sound.

In previous papers, Kwan studied the soloists. In this paper, he studies the choir. He shows that even when you are dealing with this continuous background sound, the "secret tunnel" (the reciprocity formula) still exists and works perfectly.

4. The "Recipe" for the Answer

One of the biggest mysteries in this field is: What exactly makes up the final answer?
When you calculate these moments, the result isn't just a single number; it's a sum of several different terms. Mathematicians had a "Recipe" (called the CFKRS Conjecture) that predicted exactly what these terms should look like, based on symmetry and random matrix theory. However, proving that the messy math actually matches the elegant recipe was very hard.

Kwan's breakthrough:
He proves that the "helicopter" method (the GL(3) period integral) naturally produces every single term predicted by the recipe.

He doesn't have to force the terms to fit.
He doesn't have to cancel out messy errors manually.
The terms appear naturally, like ingredients falling into a bowl when you follow a perfect recipe.

He identifies these terms using a concept called "Swaps."

Imagine you have three ingredients: A, B, and C.
A "0-swap" is just A, B, C.
A "1-swap" is swapping one ingredient (e.g., A, C, B).
A "2-swap" is swapping two.
A "3-swap" is swapping all three.

Kwan shows that the math naturally generates all these "swaps," perfectly matching the predictions of the CFKRS conjecture.

5. Why Does This Matter?

It's Cleaner: It replaces messy, brute-force calculations with a beautiful, structural understanding.
It's Deeper: It shows that the connection between these L-functions isn't a coincidence; it's built into the very geometry of the numbers.
It Opens Doors: By proving this for the "Eisenstein" case, it paves the way for solving even harder problems in the third part of the trilogy and beyond.

The Bottom Line

Chung-Hang Kwan has found a new, elegant way to look at the relationship between two complex mathematical objects. Instead of wrestling with them using old, clunky tools, he stepped back, looked at the bigger picture, and showed that the connection is a natural, symmetrical dance. He proved that the "recipe" mathematicians guessed years ago is actually the true law of the universe for these numbers.

1. Problem Statement

The paper addresses the problem of establishing a spectral reciprocity identity (a Motohashi-type formula) linking the shifted cubic moment of $GL(2)$ automorphic $L$ -functions to the shifted fourth moment of the Riemann zeta function (or $GL(1)$ $L$ -functions).

While the first paper in this series ([Kwa24]) treated the case where the $GL(3)$ form is a Hecke-Maass cusp form, this paper tackles the significantly more difficult Eisenstein case, where the $GL(3)$ form is an Eisenstein series associated with an isobaric sum $|\cdot|^{\alpha_1} \boxplus |\cdot|^{\alpha_2} \boxplus |\cdot|^{\alpha_3}$ .

The core challenges in the Eisenstein case include:

Divergence: Integrals over the non-compact domain $[G_m]$ diverge when the test function is an Eisenstein series.
Degenerate Terms: The Fourier expansion of Eisenstein series contains degenerate terms that do not vanish under unipotent integration, complicating the extraction of main terms.
Symmetry: Fully identifying the "main terms" (the asymptotic leading terms) and verifying their agreement with the CFKRS Moment Conjectures (Conrey-Farmer-Keating-Rubinstein-Snaith) requires a precise understanding of the symmetries arising from the functional equations and the Weyl group of $GL(3)$.

2. Methodology

The author employs a period integral approach within the framework of Period Reciprocity, avoiding traditional tools like the Petersson-Kuznetsov formula or Kloosterman sums.

The Core Identity: The proof relies on a "trivial" equality of two distinct period integrals for a $GL(3)$ automorphic form $\Phi$ (specifically, the Eisenstein series):
$\int_{[U_2]} \left( \int_{[G_m]} \Phi(zn) d^\times z \right) \psi(n) dn = \int_{[G_m]} \left( \int_{[U_2]} \Phi(nz) \psi(n) dn \right) d^\times z$
Here, the left-hand side (LHS) represents a $GL(3) \times GL(2)$ Rankin-Selberg period, while the right-hand side (RHS) involves a $GL(3)$ Fourier-Whittaker period.
Handling Divergence:
- The author attaches a factor $|\det|^s$ to ensure absolute convergence in a specific region ( $\text{Re}(s) \gg 1$ ).
- A key innovation is the use of the unipotent period over $[U_2]$ to eliminate the divergent degenerate components of the Eisenstein series expansion before integrating over the non-compact torus. This is achieved via a short argument (Lemma 6.5) showing that the unipotent integration annihilates the specific degenerate terms responsible for divergence.
Spectral Expansion:
- LHS: Unfolds to a spectral sum involving the shifted cubic moment of $GL(2)$ $L$ -functions (summing over cusp forms and the continuous spectrum).
- RHS: Unfolds to an integral transform involving the $GL(3)$ Whittaker function. The author uses the Fourier-Whittaker expansion of the Eisenstein series, decomposing it into non-degenerate and degenerate parts.
Analytic Continuation: The paper performs a rigorous analytic continuation of the resulting integral transforms from the region of absolute convergence to the critical strip ( $1/4 < \text{Re}(s) < 3/4$ ). This involves shifting contours in Mellin-Barnes integrals and calculating residues.
Integral Transforms: The author derives explicit formulas for the integral transform $(F_\alpha H)$ using Mellin-Barnes integrals and Gauss hypergeometric functions ( ${}_2F_1$ ). New identities (Theorem 1.4) are proven to relate these transforms to the spectral main terms.

3. Key Contributions

Exact Motohashi-Type Identity for Eisenstein Series: The paper establishes an exact formula (Theorem 1.3) linking the shifted cubic moment of $GL(2)$ $L$ -functions to the shifted fourth moment of $GL(1)$ $L$ -functions, including the continuous spectrum and all degenerate terms.
Intrinsic Derivation of Main Terms: Unlike previous approaches that required ad-hoc cancellations or complex combinatorial manipulations to find main terms, this paper derives them intrinsically from the geometry of the period integrals.
- The "0-swap" (diagonal), "1-swap", "2-swap", and "3-swap" terms (as classified by CFKRS) arise naturally from different components of the Fourier-Whittaker expansion and the poles of the analytic continuation.
Verification of CFKRS Conjectures: The paper provides a short, transparent proof that the derived main terms exactly match the predictions of the CFKRS conjectures for both the cubic moment (orthogonal symmetry) and the fourth moment (unitary symmetry).
New Mellin-Barnes Identities: The author proves three new identities (Theorem 1.4) that relate the integral transform to specific combinations of Gamma factors, effectively capturing the symmetries of the $GL(3)$ Weyl group.
Unified Framework: The method treats the cuspidal and Eisenstein cases in parallel, demonstrating that the "Period Reciprocity" framework is robust enough to handle the singularities of Eisenstein series without requiring the heavy regularization techniques used in other recent works (e.g., Nelson [Nel20]).

4. Main Results

Theorem 1.3 (The Reciprocity Formula):
For a test function $H$ and spectral parameters $\alpha$ , the shifted cubic moment $M^{(3)}_{-\alpha}(s; H)$ satisfies:
$M^{(3)}_{-\alpha}(s; H) + R_{-\alpha}(s; H) + R_{\alpha}(1-s; H) = \sum_{i=0}^3 M^i_{-\alpha}(s; H) + \frac{1}{2} \int_{(1/2)} \zeta(2s-s_0) \prod_{i=1}^3 \zeta(s_0+\alpha_i) (F_\alpha H)(s_0, s) \frac{ds_0}{2\pi i}$

LHS: The shifted cubic moment plus residual terms arising from the continuous spectrum.
RHS: A sum of main terms ( $M^0$ to $M^3$ ) corresponding to the diagonal and various "swap" terms, plus an integral transform representing the dual fourth moment.

Theorem 1.4 (Symmetries of Main Terms):
Proves that the integral transforms corresponding to the off-diagonal terms ( $M^1, M^2, M^3$ ) satisfy specific functional equations involving Gamma factors and the Weyl group action on $\alpha$ . This confirms that the structure of the main terms is dictated by the symmetries of the $GL(3)$ Weyl group.

Theorem 1.5 (Integral Transform Representation):
Expresses the integral transform $(F_\alpha H)$ explicitly in terms of the Gauss hypergeometric function ${}_2F_1$ , providing a symmetric and computable form for the dual moment.

5. Significance

Resolution of the Eisenstein Case: This work completes the picture for spectral reciprocity in the $GL(3) \times GL(2)$ setting, moving beyond the cuspidal case to include the Eisenstein spectrum, which is crucial for applications involving the continuous spectrum.
Conceptual Clarity: By deriving the main terms from the geometry of period integrals rather than combinatorial cancellations, the paper offers a deeper conceptual understanding of why the CFKRS conjectures hold. It reveals that the "swaps" correspond to specific geometric components of the Fourier expansion.
Technical Advancement: The method of using unipotent integration to regularize divergent Eisenstein periods without full geometric dissection (Atkinson dissection) is a novel and powerful technique that simplifies the analysis of spectral moments.
Applications: The established identities and the precise control over main terms are expected to have applications in:
- Proving subconvexity bounds for $L$ -functions.
- Studying simultaneous non-vanishing of $L$ -functions.
- Analyzing the distribution of values of $L$ -functions in families.

In summary, this paper represents a significant step forward in the theory of spectral reciprocity, providing a rigorous, intrinsic, and unified proof of Motohashi-type identities for Eisenstein series that aligns perfectly with modern moment conjectures.