New Bounds for Fundamental Fourier Coefficients of Siegel Modular Forms (Improved Estimates Using Iwaniec's Method)

Core Concepts

This paper presents improved bounds for the Fourier coefficients of Siegel modular forms, particularly for fundamental Fourier coefficients, by leveraging Iwaniec's method for analyzing exponential sums arising from the Petersson formula for Jacobi forms.

Abstract

• Bibliographic Information: Assing, E. (2024). New bounds for fundamental Fourier coefficients of Siegel modular forms [Preprint]. arXiv:2411.00450v1.

• Research Objective: This paper aims to improve upon existing bounds for the Fourier coefficients of Siegel modular forms, specifically focusing on fundamental Fourier coefficients.

• Methodology: The author employs Iwaniec's method to analyze the geometric side of the Petersson formula for Jacobi forms. This involves expressing the relevant Kloosterman sums as Salié sums and exploiting their explicit evaluations to extract cancellation using techniques like the Polya-Vinogradov inequality and equidistribution properties.

• Key Findings: The paper provides a new bound for the Fourier coefficients of Jacobi forms, improving upon Kohnen's 1993 result for specific ranges of the index m relative to the discriminant D. This result directly translates to an improved bound for fundamental Fourier coefficients of Siegel modular forms, surpassing previous estimates when min(T) is relatively small compared to det(T).

• Main Conclusions: By leveraging Iwaniec's method, the author successfully establishes stronger bounds for fundamental Fourier coefficients of Siegel modular forms, demonstrating the effectiveness of this approach in analyzing exponential sums arising in the context of modular forms.

• Significance: This research contributes to the study of the growth and distribution of Fourier coefficients of Siegel modular forms, a central theme in number theory with connections to L-functions and automorphic representations.

• Limitations and Future Research: The improved bounds are most effective when min(T) is significantly smaller than det(T). Further research could explore extending these techniques to achieve improvements for a wider range of min(T) values and potentially for higher genus Siegel modular forms.

Stats

k > 2 (even weight)
m ≤|D|^(7/25) (range for improvement)
min(T) ≪ det(T)^(7/25) (condition for improvement)

Quotes

"In this paper we improve Kohnen’s result if min(T) is not too large."
"This improves upon (2) as soon as min(T) ≪det(T)^(7/25)."
"Note that this improves upon Kohnen’s result stated in Theorem 1.7 for m ≤|D|^(7/25)."

Key Insights Distilled From

New bounds for fundamental Fourier coefficients of Siegel modular forms

by Edgar Assing at arxiv.org 11-04-2024

https://arxiv.org/pdf/2411.00450.pdf

Deeper Inquiries

How do these improved bounds for Fourier coefficients of Siegel modular forms impact related areas of number theory, such as the study of L-functions or the Langlands program?

Improved bounds for Fourier coefficients of Siegel modular forms have significant ramifications across various domains of number theory, particularly in the study of L-functions and the Langlands program. Here's a breakdown of their impact:

L-functions: Siegel modular forms, specifically their associated L-functions, provide a crucial link between the realm of automorphic forms and arithmetic geometry. The Fourier coefficients of these forms directly influence the analytic behavior of their L-functions. Having sharper bounds on these coefficients leads to:

Improved understanding of analytic properties: We gain better control over the growth and distribution of values of L-functions. This is essential for investigating fundamental conjectures like the Generalized Riemann Hypothesis and the Lindelöf Hypothesis in the context of these L-functions.
Stronger subconvexity bounds: These bounds are central to various analytic number-theoretic applications. Improved Fourier coefficient bounds can potentially lead to sharper subconvexity estimates for families of L-functions associated with Siegel modular forms.
Applications to equidistribution problems: The distribution of values of L-functions is intimately connected to the distribution of arithmetic objects. Improved bounds can shed light on the equidistribution of special values of L-functions, which in turn has implications for the distribution of related arithmetic objects.

Langlands Program: The Langlands program posits a profound connection between automorphic representations and Galois representations. Siegel modular forms play a crucial role in this program, and their Fourier coefficients provide a bridge between these two worlds. The improved bounds have the following implications:

Refined understanding of the correspondence: The Langlands correspondence conjectures a precise dictionary between automorphic and Galois representations. Sharper bounds on Fourier coefficients can potentially lead to a more refined understanding of this correspondence for representations associated with Siegel modular forms.
Construction of Galois representations: In some instances, the construction of Galois representations relies on knowing the existence of automorphic forms with prescribed properties, including bounds on their Fourier coefficients. Improved bounds can potentially expand the class of Galois representations that can be constructed via this automorphic approach.
In summary, the improved bounds for Fourier coefficients of Siegel modular forms provide valuable tools for tackling fundamental questions in the study of L-functions and the Langlands program. These advancements have the potential to deepen our understanding of the interplay between automorphic forms, arithmetic geometry, and representation theory.

Could alternative methods from analytic number theory, beyond Iwaniec's approach, be employed to potentially achieve even stronger bounds for these Fourier coefficients?

While Iwaniec's method has proven highly effective in obtaining strong bounds for Fourier coefficients of Siegel modular forms, exploring alternative approaches from analytic number theory holds the potential to yield even sharper estimates. Here are some avenues worth investigating:

Spectral Theory of Automorphic Forms: Delving deeper into the spectral theory of automorphic forms on higher-rank groups like Sp(2n) could provide valuable insights. This might involve:

Kuznetsov-type trace formulas: These formulas relate Fourier coefficients to sums of Kloosterman sums. Refining these formulas or finding new variants tailored to Siegel modular forms could lead to improved bounds.
Bounding triple product L-functions: The size of Fourier coefficients is related to central values of certain triple product L-functions. Obtaining strong subconvexity bounds for these L-functions would directly translate into improved Fourier coefficient bounds.

Circle Method and Exponential Sums: The circle method, a powerful tool in analytic number theory, could be employed to analyze the Fourier coefficients. This would involve:

Decomposing the coefficients: Expressing the coefficients as suitable exponential sums and then carefully analyzing these sums using methods like Poisson summation and stationary phase.
Exploiting cancellation in exponential sums: Developing new techniques or applying existing ones like Vinogradov's method or the theory of shifted convolutions to detect and exploit cancellation in the relevant exponential sums.

Geometric Methods: Leveraging the connection between Siegel modular forms and arithmetic geometry could offer alternative approaches:

Moduli spaces of abelian varieties: Siegel modular forms parametrize certain moduli spaces of abelian varieties. Geometric techniques on these moduli spaces might yield new insights into the distribution of Fourier coefficients.
p-adic methods: Exploring p-adic families of Siegel modular forms and applying p-adic analytic techniques could potentially lead to stronger bounds, especially in situations where classical methods face limitations.
It's important to note that achieving further improvements in these bounds is a challenging endeavor. The current bounds are often close to what's conjectured to be optimal. Nevertheless, exploring these alternative approaches and developing new techniques hold the promise of pushing the boundaries of our understanding and potentially achieving even stronger results.

What are the implications of these findings for understanding the distribution of modular forms and their associated arithmetic objects, such as elliptic curves or Galois representations?

The improved bounds on Fourier coefficients of Siegel modular forms have profound implications for our understanding of the distribution of modular forms themselves and their intricate connections to arithmetic objects like elliptic curves and Galois representations. Here's an exploration of these implications:

Distribution of Modular Forms:

Rarity of Siegel modular forms: Sharper bounds on Fourier coefficients provide stronger constraints on the existence of Siegel modular forms with specific properties. This suggests that forms with exceptionally large coefficients are relatively rare.
Bias in the distribution: The bounds can shed light on potential biases in the distribution of Siegel modular forms within their spaces. For instance, forms with small Fourier coefficients might be more prevalent than those with larger coefficients.

Elliptic Curves:

Congruences between elliptic curves: Siegel modular forms, particularly those associated with paramodular groups, are closely related to families of elliptic curves. Improved bounds on Fourier coefficients can lead to a better understanding of congruences between elliptic curves, which are crucial for studying the arithmetic of these curves.
Ranks of elliptic curves: The Birch and Swinnerton-Dyer conjecture predicts a deep connection between the rank of an elliptic curve and the order of vanishing of its L-function. Improved bounds on Fourier coefficients of related Siegel modular forms could potentially provide insights into the distribution of ranks in families of elliptic curves.

Galois Representations:

Distinguishing Galois representations: Fourier coefficients of Siegel modular forms are often related to traces of Frobenius elements in associated Galois representations. Sharper bounds on these coefficients can aid in distinguishing different Galois representations, which is crucial for understanding their properties and classifying them.
Sato-Tate distributions: The distribution of Fourier coefficients is connected to the Sato-Tate distributions of the corresponding families of Galois representations. Improved bounds can provide finer information about these distributions, leading to a more refined understanding of the statistical behavior of Frobenius traces.
In essence, the improved bounds on Fourier coefficients act as a magnifying glass, allowing us to perceive subtle patterns and structures within the realm of modular forms and their associated arithmetic objects. These findings have the potential to unravel deeper connections between seemingly disparate areas of number theory and provide valuable insights into the distribution and properties of fundamental arithmetic objects.