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Fourier Coefficients of Siegel--Eisenstein Series of Degree $2m$ and Weight $m+1$

This paper determines the constant term and exceptional non-zero Fourier coefficients of the Siegel–Eisenstein series $E_{m+1}^{(2m)}(Z)$ for $m\equiv1\pmod4$ and $m\geq5$ using Mizumoto's formula, thereby establishing a higher-degree analogue of the degree-two results by Kohnen, Nagaoka, and Haruki.

Original authors: Nobuki Takeda

Published 2026-06-09

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CC BY 4.0

Original authors: Nobuki Takeda

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 a vast, multi-dimensional landscape called the "Siegel Upper Half Space." In this world, mathematicians study special patterns called Siegel-Eisenstein series. You can think of these series as incredibly complex musical chords or intricate tapestries woven from numbers.

Usually, these tapestries are "holomorphic," meaning they are smooth, perfectly formed, and follow strict, predictable rules everywhere. However, this paper investigates a very specific, tricky scenario where the tapestry gets a little "rough" or "nearly holomorphic." It's like finding a knot in an otherwise perfect silk scarf.

Here is a breakdown of what the author, Nobuki Takeda, did, using everyday analogies:

1. The Setting: A Specific "Rough" Spot

The paper focuses on a specific type of mathematical object defined by two numbers:

Degree ( $n$ ): The size or dimension of the object. The author looks at objects where the size is an even number, specifically $n = 2m$ .
Weight ( $k$ ): A measure of complexity or "heaviness." The author looks at a specific weight where $k = m + 1$ .

The author is interested in a "boundary case." Imagine a bridge that is usually solid. For most weights, this bridge is perfectly stable (holomorphic). But at this specific weight ( $k = m+1$ ), the bridge is on the verge of wobbling. It's not falling apart, but it has a slight "non-holomorphic" wobble (dependence on a variable $Y$ ) that makes it harder to analyze.

2. The Goal: Decoding the "Fourier Coefficients"

To understand these complex tapestries, mathematicians break them down into their individual threads. This is called a Fourier expansion.

Think of the tapestry as a song. The Fourier coefficients are the individual notes that make up the song.
Some notes are "constant" (they don't change regardless of where you are in the landscape).
Other notes are "non-zero" (they change based on the specific shape of the tapestry).

The paper's main job is to write down the exact mathematical formula for every single one of these "notes" (coefficients) for this specific, tricky "wobbly" bridge.

3. The Tools: Mizumoto's "Master Recipe"

To decode these notes, the author uses a powerful tool developed by a mathematician named Mizumoto.

The Analogy: Imagine you have a giant, complicated machine (the Eisenstein series). Mizumoto provided a "user manual" or a "recipe" that tells you exactly how to take that machine apart and see what's inside.
Takeda uses this recipe to calculate the notes. He doesn't invent a new machine; he just applies the existing recipe to this specific, difficult case ( $m \equiv 1 \pmod 4$ ) and sees what comes out.

4. The Findings: What Did They Discover?

A. The "Constant" Note (The Background Hum)
Every tapestry has a background hum (the constant term).

The Surprise: For most sizes, this hum is just a simple "1" (a flat, unchanging note).
The Exception: The author found that for this specific "wobbly" case, the background hum actually changes depending on the shape of the tapestry ( $Y$ ). It's no longer a flat note; it's a note that swells and shrinks based on the geometry.
The Formula: The paper gives a precise formula for this swelling note. Interestingly, if the size $m$ is a certain type of number (specifically $m \equiv 3 \pmod 4$ ), the note stays flat. But for the author's target ( $m \equiv 1 \pmod 4$ ), the note gets complicated.

B. The "Non-Zero" Notes (The Melody)
These are the notes that appear when the tapestry has a specific shape (non-zero indices).

The Filter: The author realized that most potential notes actually cancel out or vanish. It's like tuning a radio; most frequencies are static.
The "Exceptional" Stations: Only a few specific "frequencies" (combinations of rank and size) actually produce sound. The paper identifies exactly which ones these are.
The "Square-Discriminant" Case: There is a special scenario where the shape of the tapestry has a "square" property. In this case, the notes behave differently, requiring a special calculation involving "poles" (mathematical spikes) and "residues" (the value left over after the spike).

C. The "Degree 6" Mystery
The paper mentions a specific case where the size is 6 (degree 6).

The Analogy: Imagine you are balancing a stack of blocks. You expect two specific blocks to add weight, but when you put them together, they magically cancel each other out, leaving the stack balanced.
The author proves that in the degree 6 case, two extra "middle" terms appear in the calculation, but they perfectly cancel each other out, leaving the final result clean.

5. The Big Picture

This paper is a "higher-degree analogue."

Previous Work: Mathematicians Kohnen and Nagaoka had already solved this puzzle for a small, 2-dimensional tapestry (degree 2).
This Paper: Takeda takes that same logic and scales it up to much larger, more complex tapestries (degree $2m$ ). He shows that the rules for the small tapestry hold true for the big ones, but with more complex "wobbles" and cancellations.

Summary

In short, the author took a known, difficult mathematical problem (analyzing a specific type of high-dimensional number pattern) and used a powerful existing formula to map out every single piece of it. He found that while the pattern is mostly stable, there are specific "wobbly" parts where the numbers change in a predictable, complex way, and he wrote down the exact instructions for calculating those changes. This extends a known solution from a small, simple case to a much larger, more general family of cases.

Technical Summary: Fourier Coefficients of Siegel–Eisenstein Series of Degree $2m$ and Weight $m+1$

Problem Statement
This paper investigates the Fourier expansion of the Siegel–Eisenstein series $E^{(n)}_k(Z)$ of degree $n = 2m$ and weight $k = m+1$ , specifically in the case where $m \equiv 1 \pmod 4$ and $m \ge 5$ . The weight $k = (n+2)/2$ represents a boundary case in the theory of Siegel modular forms. While Shimura and Weissauer established that Siegel–Eisenstein series are holomorphic for weights $k \ge (n+1)/2$ except for specific exceptional cases, the case $k = (n+2)/2 \equiv 2 \pmod 4$ is known to produce non-holomorphic dependence on the imaginary part $Y$ of the variable $Z = X + \sqrt{-1}Y$ .

The primary objective is to explicitly determine the Fourier coefficient functions $A^{(n)}_T(Y)$ for this specific weight and degree. This includes calculating the constant term and the coefficients for non-zero indices $T$ , identifying the precise nature of the non-holomorphic terms (specifically their dependence on $\sqrt{\det Y}$ ), and distinguishing between exceptional cases where poles in the underlying $L$ -functions or Siegel series interact with zeros in the gamma factors.

Methodology
The analysis relies fundamentally on the general Fourier expansion formula for Siegel–Eisenstein series established by Mizumoto. The paper proceeds by decomposing the series into contributions from Bruhat cells of varying ranks $\nu$ and analyzing the behavior of these terms as the complex parameter $s$ approaches 0 (where the series is evaluated).

Key methodological components include:

Mizumoto's Expansion: The series is expressed as a sum over $\nu$ (rank of the cell) and $\lambda$ (rank of the index matrix $T$ ). Each term involves Siegel series, Koecher–Maass zeta functions, and confluent hypergeometric functions.
Laurent Expansion at $s=0$ : Since the series is defined at $s=0$ $s = 0$ , the authors compute the finite part of the Laurent expansion for each component. This requires careful tracking of zeros and poles arising from:
- Gamma factors $\Gamma_\nu(s)$ .
- Siegel series $S_\nu(T, s)$ .
- Koecher–Maass zeta functions $\zeta^{(n)}_\nu(g, s)$ .
- The completed Riemann zeta function $\xi(s)$ .
Case Analysis by Rank:
- Constant Term ( $T=0$ ): The calculation involves summing contributions from $\nu = 0, 1, 2, 2m-1, 2m$ . The authors utilize functional equations and residue formulas for Epstein zeta functions to simplify the sum.
- Non-zero Indices ( $T \neq 0$ ): The analysis distinguishes between positive semi-definite ( $T \ge 0$ ) and non-positive semi-definite ( $T \not\ge 0$ ) indices.
- Exceptional Pairs: The authors define specific index sets $J^{(n)}_\lambda$ where the scalar factors in Mizumoto's formula have non-positive order at $s=0$ . These are the only terms that contribute to the final Fourier coefficients.
- Square-Discriminant Poles: Special attention is paid to cases where the Siegel series has a simple pole (occurring when the rank $\lambda$ is even and the discriminant of the associated matrix is a square). The interaction between these poles and the zeros of the scalar factors determines whether a non-zero coefficient exists.

Key Contributions and Results

Constant Term Formula (Theorem 4.1 and 6.1):
The paper derives an explicit formula for the constant term $A^{(2m)}_0(Y)$ .
- For $m \equiv 3 \pmod 4$ , the constant term is simply $1$, consistent with the holomorphicity of the series in that case.
- For $m \equiv 1 \pmod 4$ , the constant term exhibits non-trivial dependence on $Y$ :
  $A^{(2m)}_0(Y) = 1 + \frac{(\epsilon_m - 1)\pi^{m+1}}{\zeta(m+1)\zeta(2m)\sqrt{\det Y}} \left( \frac{\zeta(m)}{2^{m+1}m!} + \frac{(m-2)!\zeta(m-1)}{(2m)!} \zeta^{(2m)}_1(Y, m-1) \right)$
  where $\epsilon_m = (-1)^{(m+1)/2}$ and $\zeta^{(2m)}_1(Y, m-1)$ is the Koecher–Maass zeta function understood via meromorphic continuation. This result generalizes the degree-two formulas of Kohnen, Nagaoka, and Haruki to higher degrees.
Non-zero Index Terms (Propositions 5.3 and 5.5):
The paper provides a complete characterization of the coefficients $A^{(n)}_T(Y)$ for $T \neq 0$ .
- Positive Semi-definite $T$ : The coefficient is a finite sum over a specific set of ranks $\nu \in J^{(n)}_\lambda$ . The terms are expressed using Siegel series $S_\lambda(h, u)$ and values of the confluent hypergeometric function (or its derivatives) at specific points.
- Non-positive Semi-definite $T$ : The paper identifies that non-zero coefficients only arise for specific pairs of rank $\lambda$ and cell rank $\nu$ , determined by the signature $(p, q)$ of the matrix $h$ and the parity of the discriminant.
- Exceptional Cases: The authors explicitly handle cases where the scalar factor has a zero that cancels a pole in the Siegel series (specifically for $(\lambda, \nu) = (n-2, n)$ and square-discriminant cases). In these instances, the coefficients involve derivatives of the Siegel series or the associated $L$ -functions.
Degree 6 Cancellation (Appendix A):
The paper notes that for $m=3$ (degree 6), additional middle-rank terms ( $\nu=3, 4$ ) appear in the constant term calculation. However, it is shown that these terms cancel each other out, allowing the general formula for $m \ge 5$ to hold uniformly for the constant term.

Significance
The paper claims to provide a higher-degree analogue of the degree-two formulas for the exceptional weight $k=2$ (where $n=2$ ). By utilizing Mizumoto's general expansion formula, the author unifies the treatment of the constant term and non-zero indices for degree $n=2m$ with $m \equiv 1 \pmod 4$ . The work clarifies the structure of the non-holomorphic dependence on $Y$ in this boundary weight, showing that it arises from specific interactions between the poles of Siegel series and the zeros of gamma and zeta factors. The results extend the understanding of nearly holomorphic modular forms and the Fourier expansions of Eisenstein series in the critical range of weights.