[Paper Review] Weights of the mod $p$ kernel of the theta operators

Siegfried Boecherer, Toshiyuki Kikuta+1 more|arXiv (Cornell University)|Jun 21, 2016

Advanced Algebra and Geometry25 references3 citations

TL;DR

This paper investigates the weights of Siegel modular forms in the mod $p$ kernel of generalized theta operators $\Theta^{[j]}$, establishing that for small weights relative to $p$, the weight must be divisible by $p$ under certain conditions. It introduces a new operator $A^{(j)}(M)$ to construct elements in the kernel from arbitrary modular forms, and provides explicit examples and filtrations of such forms for various weights and primes.

ABSTRACT

We give some relations between the weights and the prime $p$ of elements of the mod $p$ kernel of the generalized theta operator $Θ^{[j]}$. In order to construct examples of the mod $p$ kernel of $Θ^{[j]}$ from any modular form, we introduce new operators $A^{(j)}(M)$ and show the modularity of $F|A^{(j)}(M)$ when $F$ is a modular form. Finally, we give some examples of the mod $p$ kernel of $Θ^{[j]}$ and the filtrations of some of them.

Motivation & Objective

• To determine the necessary conditions on the weight and prime $p$ for elements to lie in the mod $p$ kernel of generalized theta operators $\Theta^{[j]}$ in Siegel modular forms.
• To extend the classical theory of Ramanujan's $\theta$-operator to higher-degree Siegel modular forms and their mod $p$ kernels.
• To construct new elements in the mod $p$ kernel of $\Theta^{[j]}$ from arbitrary modular forms using a novel operator $A^{(j)}(M)$.
• To compute and tabulate the filtrations of specific elements in the mod $p$ kernel of $\Theta^{[j]}$ for various weights and primes.
• To generalize previous results on congruences and filtrations in the context of Siegel modular forms modulo $p$.

Proposed method

• Introduce a new operator $A^{(j)}(M)$ that acts on modular forms to produce elements in the mod $p$ kernel of $\Theta^{[j]}$, ensuring modularity of the output.
• Use the generalized theta operator $\Theta^{[j]}$, defined as $\sum_T T a_F(T) q^T$ for $F = \sum_T a_F(T) q^T$, to analyze the kernel modulo $p$.
• Apply the theory of Fourier expansions and $p$-adic properties of Siegel modular forms to study the structure of the kernel.
• Leverage known results on Klingen-Eisenstein series and cusp forms to construct explicit examples in the kernel.
• Compute filtrations of kernel elements using the $p$-adic filtration theory, with detailed tables for weights up to 100 and primes below 80.
• Utilize the action of the symplectic group and the slash operator to define modular invariance and ensure the modularity of $F|A^{(j)}(M)$.

Experimental results

Research questions

• RQ1 What conditions on the weight $k$ and prime $p$ are necessary for a Siegel modular form to lie in the mod $p$ kernel of $\Theta^{[j]}$?
• RQ2 Can new elements in the mod $p$ kernel of $\Theta^{[j]}$ be systematically constructed from arbitrary modular forms?
• RQ3 What are the explicit filtrations of elements in the mod $p$ kernel of $\Theta^{[j]}$ for small weights and various primes?
• RQ4 How do the filtrations of kernel elements behave as the weight and prime vary, especially in relation to $p$-divisibility?
• RQ5 To what extent do the properties of the classical $\theta$-operator generalize to the higher-rank $\Theta^{[j]}$ operators in the Siegel modular setting?

Key findings

• For small weights relative to $p$, the weight $k$ of any element in the mod $p$ kernel of $\Theta^{[j]}$ must be divisible by $p$, generalizing Serre and Katz's results to higher-degree Siegel modular forms.
• The operator $A^{(j)}(M)$ successfully generates elements in the mod $p$ kernel of $\Theta^{[j]}$ from any modular form $F$, and $F|A^{(j)}(M)$ remains modular.
• Explicit examples of elements in the mod $p$ kernel of $\Theta^{[j]}$ are provided, including for weights $k = 12, 16, 24, 30, 48, 54, 58, 60$, and primes $p = 23, 31, 47, 59, 73$, among others.
• Filtrations of kernel elements are computed and tabulated; for instance, $\omega_1(F) = 1$ for $F$ of weight 24 and $p=23$, and $\omega_1(F) = 1$ for weight 54 and $p=59$.
• The kernel of $\Theta^{[2]}$ contains non-zero forms for weights $k = 18, 24, 28, 30, 38, 42, 48, 50, 54, 58, 60, 66, 68, 72, 74, 78, 80, 83, 84, 88, 90, 92, 93, 96, 98$ modulo $p < 80$.
• For $\Theta^{[1]}$, the paper confirms that the filtration $\omega_1(F)$ is 1 for forms of weight 24 and $p=23$, and 1 for weight 54 and $p=59$, indicating minimal $p$-adic valuation.