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Fundamental Fourier coefficients of Siegel modular forms of higher degrees and levels

This paper proves that for any Siegel modular form of degree $n$ and odd level $N$ with at least one Fourier coefficient $A(F, T_0)$ where the last diagonal entry of $T_0$ is coprime to $N$ , there exist infinitely many fundamental discriminants $m$ for which the Fourier coefficient $A(F, T)$ is non-zero, a result that is used to provide an unconditional proof for the algebraicity of certain critical values of spinor $L$ -functions for $\mathrm{GSp}(3)$ .

Original authors: Pramath Anamby, Soumya Das

Published 2026-02-10

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: Pramath Anamby, 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 master chef, and you have a secret, legendary recipe for a complex, multi-layered cake (this is the Siegel Modular Form). This cake is so sophisticated that it isn't just made of flour and sugar; it’s made of mathematical "ingredients" called Fourier coefficients.

Each ingredient is indexed by a specific "shape" or "matrix" (the $T$ ). To truly understand if a recipe is authentic or to unlock its deepest secrets (like its L-functions, which are like the "nutritional profile" of the cake), you don't just need to know that the cake exists. You need to know if it contains specific, high-quality, "pure" ingredients.

The Problem: Finding the "Pure" Ingredients

In mathematics, some ingredients are "diluted" or "messy" (these are non-fundamental or non-primitive coefficients). They are like using pre-mixed, processed flour. To prove certain deep mathematical theories, scientists need to prove that the recipe must contain "Fundamental" ingredients—think of these as organic, single-origin, stone-ground flour that hasn't been mixed with anything else.

For a long time, mathematicians knew these pure ingredients existed for very simple recipes (Level 1). But when the recipes got complicated (Higher Degrees and Higher Levels), it became a nightmare to prove they were there. It was like trying to prove that a massive, industrial-scale bakery definitely uses organic flour, even though they have thousands of different mixing bowls and complex machinery.

The Breakthrough: The Mathematical "Sieve"

Pramath Anamby and Soumya Das have created a way to prove that these "pure" ingredients (fundamental, square-free, or even prime discriminants) always exist in these complex recipes.

They used a technique called Induction. Imagine you have a giant, 10-layer cake. You want to prove the 10th layer has pure ingredients. You can't do it all at once. So, you prove it for a 1-layer cake, then use that to prove it for a 2-layer cake, and so on, until you reach 10.

However, there was a "glitch" in the previous methods: when you tried to move from one layer to the next, the "purity" of the ingredients would get lost in the transition (the "level" would inflate or the "symmetry" would break).

The authors solved this by:

Creating a specialized "Mixing Bowl": They defined a very specific type of mathematical group (the $\Gamma^{\pm}(N, L)$ group) that preserves the "purity" of the ingredients as they move between layers.
The "Symmetry" Trick: They added a special "sign" component (the $E$ group) to ensure that the mathematical equations stayed balanced, preventing the "purity" from washing away during the induction process.

Why Does This Matter? (The "So What?")

This isn't just about cake; it's about the "DNA" of mathematics.

Unlocking the L-functions: There is a famous set of conjectures (Deligne’s conjectures) that predict how certain mathematical values behave. Previously, these predictions were only "conditional"—meaning they were only true if we could prove these pure ingredients existed. This paper turns those "ifs" into "certainties." It makes the math unconditional.
The Prime Connection: For certain types of these forms, the authors proved that you can find ingredients that are not just pure, but Prime. In the world of numbers, primes are the "atoms." Proving that these forms are built from "prime atoms" is a massive win for number theory.
Quadratic Forms: It also helps in understanding how one geometric shape can be "represented" by another, which is a fundamental question in how we understand space and structure.

Summary in a Sentence

The authors have provided a master key that proves even the most complex, high-level mathematical "recipes" are built from the purest, most fundamental "atoms" of number theory, finally turning many mathematical "maybe's" into "definitely's."

This paper, titled "Fundamental Fourier Coefficients of Siegel Modular Forms of Higher Degrees and Levels" by Pramath Anamby and Soumya Das, addresses a central problem in the analytic and arithmetic theory of Siegel modular forms: the existence and distribution of Fourier coefficients indexed by matrices with "arithmetically interesting" properties (specifically, fundamental discriminants).

Below is a detailed technical summary of the work.

1. The Problem

The authors investigate the non-vanishing of Fourier coefficients $A(F, T)$ of a Siegel modular form $F$ of degree $n$ and level $N$ . In the theory of $L$ -functions, it is often necessary to know that a modular form has non-zero coefficients where the index $T$ has a fundamental discriminant (or is prime, or square-free).

While such results were known for level $N=1$ (degree 2 and 3), extending these results to arbitrary odd levels $N > 1$ and higher degrees $n \geq 3$ presents significant technical hurdles. Specifically, the "oldform" structure of higher-level congruence subgroups and the complexities of the Fourier-Jacobi expansion make it difficult to ensure that the "fundamental" property is preserved during inductive processes.

2. Methodology

The authors employ a sophisticated inductive approach based on the degree of the modular form. The proof structure is organized into several technical stages:

Selection of Congruence Subgroups: To make the induction work, the authors do not work directly with the standard Hecke congruence subgroups $\Gamma_0^{(n)}(N)$ . Instead, they define and work with a specific intermediate group $\Gamma_{\pm}^{(n)}(N, L)$ , where $L$ is the square-free part of $N$ . This group is a semi-direct product involving a sign-matrix group $E$ , which ensures a "symmetry" property in the Fourier-Jacobi coefficients necessary for Gauss sum calculations.
Fourier-Jacobi (FJ) Models: The core of the induction involves descending from degree $n$ to $n-1$ . The authors use the Fourier-Jacobi expansion to relate the Siegel modular form to vector-valued Jacobi forms.
Theta Decomposition: They utilize the theta decomposition of Jacobi forms to reduce the problem to the study of "theta components" $h_\mu$ . A major part of the paper is dedicated to proving that if a Jacobi form is non-zero, at least one of its primitive theta components must be non-zero.
Gauss Sums and Maximal Rank Arguments: To prove the non-vanishing of these components, the authors perform intricate calculations involving higher-dimensional quadratic Gauss sums. They demonstrate that certain matrices arising from these sums have maximal rank, which allows them to "untangle" the Fourier coefficients and isolate those with fundamental discriminants.

3. Key Contributions and Results

A. Main Existence Theorems

Theorem 1.1 (Quantitative Result): For a non-zero Siegel modular form $F$ of degree $n$ and odd level $N$ , if there exists at least one non-zero coefficient $A(F, T_0)$ with $(T_0(n,n), N)=1$ , then there are infinitely many $T$ such that $A(F, T) \neq 0$ and $d(T)$ is odd and square-free (fundamental). They provide quantitative lower bounds for the number of such coefficients up to $X$ .
Theorem 1.3 (Refined Induction): They prove a stronger version where the index $T$ is not only fundamental but also diagonal modulo $L$ . This is a crucial technical stepping stone for the induction.
Prime Discriminants (Odd Degrees): For odd degrees $n$ , the authors prove a stronger result: there exist infinitely many $T$ such that $d(T)$ is an odd prime.

B. Characterization of Fundamental Coefficients

Theorem 1.4: They establish a criterion for the existence of fundamental coefficients: a form has a non-zero fundamental Fourier coefficient if and only if it has a non-zero "primitive" Fourier coefficient. This simplifies the search for fundamental coefficients to a check for primitivity.

C. Unconditionality of $L$ -function Results

Theorem 1.5: A major consequence is that the algebraicity of certain critical values of spinor $L$ -functions for holomorphic newforms on $\Gamma_0^{(3)}(N)$ (previously conditional on the existence of certain quaternion algebra orders) is now unconditional.

4. Significance

The paper is significant for several reasons:

Bridging Theory and Application: It connects the internal Fourier structure of Siegel modular forms to the analytic properties of $L$ -functions and the theory of quadratic forms (representation numbers).
Overcoming the "Level" Barrier: By carefully choosing the congruence subgroups and handling the "lattice-level" of Jacobi forms, the authors successfully bypassed the obstacles that prevented previous researchers from extending level-one results to higher levels.
Mathematical Completeness: The work settles long-standing questions regarding the existence of fundamental coefficients for higher-degree forms at arbitrary odd levels, providing a robust framework that can likely be extended to other groups (like paramodular or unitary groups) in the future.