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Moments and joint nonvanishing of symplectic $L$ -functions

This paper establishes an asymptotic formula for a moment involving spinor and standard $L$ -functions of holomorphic Siegel cusp forms of degree two and large weight, which is then applied to prove simultaneous non-vanishing results and derive lower bounds for second moments.

Original authors: Valentin Blomer, Soumya Das

Published 2026-04-24

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Valentin Blomer, Soumya Das

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 solve a mystery about a vast, invisible city made entirely of numbers. This city is called the world of Symplectic L-functions.

In this paper, two mathematicians, Valentin Blomer and Soumya Das, act as master cartographers. Their goal? To draw a map of a specific neighborhood in this city that no one has ever successfully charted before.

Here is the story of their journey, explained without the heavy math jargon.

1. The Mystery: The "Ghost" Numbers

In the world of numbers, there are special functions called L-functions. Think of these as "musical instruments" that play a specific tune based on a number.

Some instruments are simple and well-known (like a flute).
Others are complex, massive orchestras (like a symphony).

The authors are studying two specific instruments attached to a special type of mathematical object called a Siegel cusp form (think of this as a very complex, multi-dimensional wave pattern).

Instrument A (Spinor L-function): A 4-note chord. We know a lot about this one.
Instrument B (Standard L-function): A 5-note chord. This one is much more complicated, like a chaotic jazz improvisation.

The Big Question: If you play these instruments at a specific, critical moment (mathematicians call this $s = 1/2$ ), do they make a sound (is the value non-zero), or do they go silent?

For decades, mathematicians have been able to predict the average volume of Instrument A. But for Instrument B, it's been a black box. No one knew if the average volume was loud, quiet, or zero.

2. The Tool: The "Grand Piano" Formula

To listen to these instruments, the authors use a special tool called the Kitaoka Formula.

The Analogy: Imagine you have a giant piano with thousands of keys (representing thousands of different mathematical waves). You want to know the average sound of a specific note played across all these keys.
Usually, you could just press the keys and listen. But in this math world, the keys are hidden, and the sound is distorted by "noise" (mathematical terms called Kloosterman sums and Bessel functions).

The authors' job is to filter out the noise to hear the true signal.

3. The Journey: Three Obstacles

The authors break their investigation into three parts, like climbing a mountain with three distinct camps.

Camp 1: The Main Peak (The Diagonal Term)

This is the "easy" part of the climb. It's the direct path.

What happens: They calculate the average sound directly.
The Result: They find a clear, loud signal! The average volume grows as the "weight" of the instruments increases. Specifically, they prove that the average sound is roughly proportional to the logarithm of the weight (a slow, steady growth).
Why it matters: This proves that, on average, these instruments do make a sound. They aren't silent.

Camp 2: The Foggy Ridge (The Rank One Term)

This is where things get tricky. The path is covered in fog (mathematical noise).

The Problem: The noise here is so strong that standard listening devices (trivial bounds) can't hear the signal. It's like trying to hear a whisper in a hurricane.
The Solution: The authors invent a new "noise-canceling headphone" technique. They use a method called Poisson Summation (imagine rearranging the fog into a pattern so the wind stops blowing).
The Result: They prove that this noise is actually much quieter than anyone thought. It's negligible. The signal from Camp 1 stands clear.

Camp 3: The Deep Cave (The Rank Two Term)

This is the hardest part. It's a deep cave where the geometry of the space itself is twisted.

The Problem: Here, the "noise" depends on how close the mathematical shapes are to being perfect squares (orthogonal matrices). If they are close to perfect, the noise cancels out the signal completely.
The Solution: The authors realize they need to measure exactly how close these shapes are to being perfect. They create a new coordinate system (a new way of looking at the cave) to separate the "long" variables from the "short" ones.
The Result: They show that even in the deepest, most twisted part of the cave, the noise is still under control. The signal survives.

4. The Treasure: What Did They Find?

After conquering the mountain, they bring back two major discoveries:

The "Non-Vanishing" Proof:
They proved that for large enough weights, there is always at least one instrument in the orchestra that is playing loudly.
- In plain English: It is impossible for all of these complex mathematical functions to be zero at the same time. At least one of them is "alive."
The Lower Bound:
They proved that the "energy" (the square of the volume) of these instruments is huge.
- In plain English: Not only do they make a sound, but they make a loud sound. This gives mathematicians a solid foundation to build future theories on.

5. Why Should You Care?

You might ask, "Who cares about 5-note chords in a 4D city?"

The Riemann Hypothesis Connection: These L-functions are related to the most famous unsolved problem in math (the Riemann Hypothesis). Understanding their "volume" helps us understand the distribution of prime numbers (the building blocks of all numbers).
The "Non-Vanishing" Guarantee: In cryptography and coding theory, knowing that a function doesn't vanish (go to zero) is crucial. If a code relies on these numbers, and they turn out to be zero, the code breaks. This paper guarantees the code is safe.
The Method: The techniques they used to "cancel the noise" are so clever that other mathematicians can now use them to solve different, unrelated mysteries in the number world.

Summary

Blomer and Das took a chaotic, noisy mathematical problem that had stumped experts for years. They built a new set of tools to filter out the noise, climbed the three hardest parts of the problem, and proved that the "Standard L-function" is not a ghost—it has a real, measurable, and non-zero presence. They didn't just find a needle in a haystack; they proved the haystack is full of needles.

1. Problem Statement and Context

The paper addresses the asymptotic evaluation of moments of $L$ -functions associated with holomorphic Siegel cusp forms of degree 2 (genus 2) for the symplectic group $Sp(4, \mathbb{Z})$ . Specifically, the authors investigate a mixed moment involving two distinct $L$ -functions attached to a Siegel cusp form $F$ of weight $k$ :

The Spinor L-function $L(s, F)$ : Degree 4, conductor $k^2$ .
The Standard L-function $D(s, F)$ : Degree 5, conductor $k^4$ .

While moments of spinor $L$ -functions have been studied previously (e.g., in [Bl]), the standard $L$ -function is significantly more complex due to its Dirichlet series expansion involving matrix Fourier coefficients rather than just scalar coefficients. Prior to this work, no moment results existed for the standard $L$ -function in this context.

The primary goal is to compute the first moment of the standard $L$ -function at the central point $s=1/2$ , averaged over an eigenbasis of Siegel cusp forms with harmonic weights. This leads to a joint non-vanishing problem: determining if there exist forms where both the spinor and standard $L$ -functions (and their twists) are simultaneously non-zero at the central point.

2. Methodology

The authors employ a sophisticated analytic number theory approach combining spectral summation formulas, approximate functional equations, and advanced exponential sum estimates.

A. Spectral Summation (Kitaoka Formula)

The core tool is Kitaoka's formula, a higher-rank analogue of the Petersson trace formula for $Sp(4, \mathbb{Z})$ . This formula allows the authors to average products of Fourier coefficients $a_F(M)a_F(Q)$ over the space of cusp forms.

The formula decomposes the spectral sum into three parts:
1. Diagonal term: Yields the main term.
2. Rank one term: Involves matrix Kloosterman sums and Bessel functions.
3. Rank two term: Involves more complex matrix Kloosterman sums and Bessel functions.

B. Approximate Functional Equation

The authors insert an approximate functional equation for $D(1/2, F)$ into the spectral sum. This converts the $L$ -value into a Dirichlet series involving Fourier coefficients $a_F(MM^\top)$ weighted by a smooth test function $W(x)$ .

C. Analysis of Error Terms (Rank One and Rank Two)

The bulk of the technical difficulty lies in bounding the off-diagonal terms (Rank 1 and Rank 2) to ensure they are smaller than the main term (saving a power of $k$ ).

Rank One Term:
- The authors analyze matrix Kloosterman sums $H^\pm(P, S; c)$ .
- They prove Lemma 1, establishing a square-root cancellation bound for these sums, $|H^\pm| \ll c^{1+\epsilon}$ .
- Crucially, trivial bounds are insufficient. The authors apply Poisson summation to the matrix variables to exploit cancellation in the exponential sums, achieving a power-saving error term.
Rank Two Term (The Most Difficult Part):
- This term involves matrix Bessel functions $J_\ell$ and Kloosterman sums $K(Q, T; C)$ .
- Coordinate Transformation: The authors introduce a novel coordinate system to measure the "distance" of the matrix argument $NJ^{-1}$ from being orthogonal. They decompose the space of matrices into subspaces $W_+$ and $W_-$ where the matrix is "close" to orthogonal.
- Oscillation Analysis: They observe that the matrix Bessel function has reduced oscillation when the matrix is close to orthogonal. By quantifying this proximity, they can apply stationary phase arguments and Poisson summation selectively (summing over "long" variables but not "short" ones).
- Uniform Asymptotics: They utilize Olver's uniform asymptotic expansion for Bessel functions to handle the transition regions where the argument is large.

3. Key Contributions and Results

Main Theorem (Theorem 1)

The authors derive an asymptotic formula for the mixed moment:
$M_k := \sum_{F \in \mathcal{B}_k^{(2)}} \omega_F D(1/2, F) = 2L(3/2, \chi_{-4}) \log k + c_0 + O(k^{-\eta})$
where $\omega_F$ are harmonic weights, $\chi_{-4}$ is the non-trivial Dirichlet character modulo 4, and $c_0$ is an explicit constant involving Euler's constant $\gamma$ and derivatives of $L$ -functions.

The error term $O(k^{-\eta})$ (for $\eta < 1/4$ ) represents a power-saving over the trivial bound, which is a significant achievement in higher-rank automorphic forms.
The contribution of Saito-Kurokawa lifts (lifts from elliptic modular forms) is shown to be negligible.

Corollary 2: Joint Non-Vanishing

A direct consequence of the main theorem is a simultaneous non-vanishing statement. For sufficiently large weight $k$ , there exists a Siegel cusp form $F$ such that:

$L(1/2, F) \neq 0$ (Spinor)
$L(1/2, F \times \chi_{-4}) \neq 0$ (Twisted Spinor)
$D(1/2, F) \neq 0$ (Standard)
Furthermore, $D(1/2, F) > 0$ .

Note: The positivity of $D(1/2, F)$ relies on the Generalized Riemann Hypothesis (GRH) and the intermediate value theorem, as unconditional period formulas are not yet available for this specific case.

Corollary 3: Lower Bound for Second Moment

Using the Cauchy-Schwarz inequality and the main theorem, the authors establish a lower bound for the second moment of the standard $L$ -function:
$\sum_{F \in \tilde{\mathcal{B}}_k^{(2)}} \frac{|D(1/2, F)|^2}{L(1, \pi_F, \text{Ad})} \gg k^3$
This confirms the expected order of magnitude for the second moment in the symplectic family.

4. Significance and Impact

First Moment for Standard L-functions: This is the first result providing an asymptotic formula for the first moment of the standard $L$ -function of degree 5 Siegel cusp forms. This fills a major gap in the literature, as the standard $L$ -function is structurally more complex than the spinor $L$ -function.
Higher-Rank Techniques: The paper develops and refines techniques for handling matrix Kloosterman sums and matrix Bessel functions in the context of $Sp(4)$. The specific analysis of the "orthogonality" of matrices in the Rank 2 term is a novel contribution to the spectral theory of automorphic forms.
Non-Vanishing Applications: The results provide strong evidence for the non-vanishing of central $L$ -values in the symplectic family, which has implications for the arithmetic of Siegel modular forms and the distribution of zeros of $L$ -functions (low-lying zeros).
Methodological Benchmark: The detailed analysis of the Rank 2 term, particularly the use of coordinate transformations to separate oscillatory and non-oscillatory regimes, serves as a template for future investigations into higher-rank moments and subconvexity problems.

In summary, this paper represents a major advancement in the analytic theory of Siegel modular forms, successfully bridging the gap between the well-understood spinor $L$ -functions and the more elusive standard $L$ -functions through rigorous spectral analysis and exponential sum estimation.