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An algorithm to compute upper bounds of dimensions for Siegel Modular Forms of Prime Level and Arbitrary Nebentypus

Debargha Banerjee, Dron Airon, Pranjal Vishwakarma, Ronit Debnath

TL;DR

This work develops and implements an algorithm to bound the dimensions of genus-2 Siegel modular form spaces S_2^k(Γ_0^{(2)}(l), χ) at prime levels with arbitrary nebentypus, including low weights k ≤ 4. Building on Poor–Yuen restriction maps, it uses dyadic-trace bounds to identify determining coefficients, and computes their images as elliptic cusp forms to derive explicit upper bounds; nontrivial characters are accommodated via compatibility with diamond and Atkin–Lehner operations. The paper provides concrete Java programs, applies the method to examples (level 9 with trivial χ and level 13 with χ of order 3) demonstrating bounds 6 and 3 respectively, and also offers lower bounds via Saito–Kurokawa and Jacobi correspondences. Together, these results advance dimension theory for Siegel modular forms in low weight and nontrivial character settings and suggest avenues for Sturm-type results and generalizations.

Abstract

We describe an algorithmic method to determine the image of restriction maps for Siegel modular forms with \textit{arbitrary} characters and arbitrary weight. A program has been implemented in the mathematical software \texttt{Java} to compute the Fourier expansion of the image of these restriction maps for Siegel modular forms of genus two. This approach allows us to compute an upper bound for the space of Siegel modular forms with {\it non-trivial} character (which has not been previously known) and arbitrary weights (including low weight $k \leq 4$).

An algorithm to compute upper bounds of dimensions for Siegel Modular Forms of Prime Level and Arbitrary Nebentypus

TL;DR

This work develops and implements an algorithm to bound the dimensions of genus-2 Siegel modular form spaces S_2^k(Γ_0^{(2)}(l), χ) at prime levels with arbitrary nebentypus, including low weights k ≤ 4. Building on Poor–Yuen restriction maps, it uses dyadic-trace bounds to identify determining coefficients, and computes their images as elliptic cusp forms to derive explicit upper bounds; nontrivial characters are accommodated via compatibility with diamond and Atkin–Lehner operations. The paper provides concrete Java programs, applies the method to examples (level 9 with trivial χ and level 13 with χ of order 3) demonstrating bounds 6 and 3 respectively, and also offers lower bounds via Saito–Kurokawa and Jacobi correspondences. Together, these results advance dimension theory for Siegel modular forms in low weight and nontrivial character settings and suggest avenues for Sturm-type results and generalizations.

Abstract

We describe an algorithmic method to determine the image of restriction maps for Siegel modular forms with \textit{arbitrary} characters and arbitrary weight. A program has been implemented in the mathematical software \texttt{Java} to compute the Fourier expansion of the image of these restriction maps for Siegel modular forms of genus two. This approach allows us to compute an upper bound for the space of Siegel modular forms with {\it non-trivial} character (which has not been previously known) and arbitrary weights (including low weight ).

Paper Structure

This paper contains 15 sections, 6 theorems, 81 equations.

Table of Contents

  1. Introduction
  2. Notation
  3. The Restriction maps for trivial nebentypus
  4. Upper Bound of the Dyadic Traces of the Set of Determining Coefficients
  5. Program for the expansion
  6. A Java Program to Compute the Set of Determining Coefficients
  7. A Java program to compute the restriction map
  8. The Restriction maps for non-trivial nebentypus
  9. Proof of the Main Theorem
  10. Application of the Algorithm to compute the Upper bound.
  11. Siegel Modular forms of a prime power level with trivial nebentypus
  12. Seigel Modular form with nebentypus
  13. Lower Bound
  14. Lower bound for Lg
  15. Conclusion and Future Work

Key Result

Theorem 1.1

There exists an algorithm to compute the Fourier coefficients of the elliptic cusp forms that appear in the images of the restriction maps for prime level, arbitrary weight and arbitrary nebentypus. The above algorithm can be applied to determine an explicit upper bound for the dimensions of the spa for arbitrary weights $k$ (including low weight $k \leq 4$), $l$ is a prime and $\chi$ is a non-tr

Theorems & Definitions (13)

  • Theorem 1.1
  • Definition 3.1: Set of Determining Coefficients
  • Lemma 3.2
  • proof
  • Lemma 4.1
  • proof
  • Proposition 4.2
  • proof
  • Proposition 4.3
  • proof
  • ...and 3 more