[Paper Review] Note on mod p property of Hermitian modular forms

Toshiyuki Kikuta, Shōyū Nagaoka|arXiv (Cornell University)|Jan 14, 2016

Advanced Algebra and Geometry14 references3 citations

TL;DR

This paper investigates the mod p kernel of the theta operator for Hermitian modular forms of degree 2, extending prior work on Siegel modular forms. It establishes that certain Eisenstein series, theta series attached to the Hermitian Leech lattice, and Hermitian theta constants satisfy Θ(F) ≡ 0 mod p for specific primes p, using Sturm's bound and properties of the theta operator's action on modular forms.

ABSTRACT

The mod $p$ kernel of the theta operator is the set of modular forms whose image of the theta operator is congruent to zero modulo a prime $p$. In the case of Siegel modular forms, the authors found interesting examples of such modular forms. For example, Igusa's odd weight cusp form is an element of mod 23 kernel of the theta operator. In this paper, we give some examples which represent elements in the mod $p$ kernel of the theta operator in the case of Hermitian modular forms of degree 2.

Motivation & Objective

• To extend the notion of the mod p kernel of the theta operator from Siegel to Hermitian modular forms of degree 2.
• To identify explicit examples of Hermitian modular forms satisfying Θ(F) ≡ 0 mod p.
• To establish congruence results for Eisenstein series, theta series of unimodular lattices, and Hermitian theta constants under the theta operator modulo p.
• To apply Sturm's bound and known transformation properties of the theta operator to verify congruences in the mod p setting.

Proposed method

• The theta operator is defined on Hermitian modular forms via Θ(F) = ∑ det(T)·a(T)q^T for Fourier expansion F = ∑ a(T)q^T.
• The paper uses the known result that the image of a weight k modular form under the theta operator is congruent to a cusp form of weight k + p + 1 modulo p.
• It applies Sturm's bound (Corollary 2.6) to verify congruences by checking finitely many Fourier coefficients.
• The authors analyze Eisenstein series E_{k, K}^{(2)} for imaginary quadratic fields K with class number one.
• They compute Fourier coefficients of theta series associated with the Hermitian Leech lattice over the Gaussian integers and verify congruences modulo 11.
• For Hermitian theta constants, they consider ψ_{4k} = (1/4)∑_{m∈ℰ} θ_m^{4k} and verify Θ(ψ_8) ≡ 0 mod 7 and Θ(ψ_12) ≡ 0 mod 11.

Experimental results

Research questions

• RQ1 Which Hermitian modular forms of degree 2 lie in the mod p kernel of the theta operator for prime p?
• RQ2 Does the weight p+1 Hermitian Eisenstein series E_{p+1, K}^{(2)} satisfy Θ(E_{p+1, K}^{(2)}) ≡ 0 mod p when the class number of K is one?
• RQ3 Is the theta series of the Hermitian Leech lattice congruent to zero modulo 11 under the theta operator?
• RQ4 Do the Hermitian theta constants ψ_8 and ψ_12 satisfy Θ(ψ_8) ≡ 0 mod 7 and Θ(ψ_12) ≡ 0 mod 11?
• RQ5 Can the Sturm bound be effectively used to verify mod p congruences for Hermitian modular forms?

Key findings

• The Hermitian Eisenstein series F_{p+1, K} satisfies Θ(F_{p+1, K}) ≡ 0 mod p under the condition that the class number of K is one.
• The theta series of the Hermitian Leech lattice over the Gaussian integers satisfies Θ(ϑ_{ℒ_ℂ}^{(2)}) ≡ 0 mod 11.
• The Hermitian modular form ψ_8 = (1/4)∑_{m∈ℰ} θ_m^8 satisfies Θ(ψ_8) ≡ 0 mod 7.
• The Hermitian modular form ψ_12 = (1/4)∑_{m∈ℰ} θ_m^{12} satisfies Θ(ψ_12) ≡ 0 mod 11.
• The Fourier coefficients of ϑ_{ℒ_ℂ}^{(2)} were computed up to trace 6, and all non-zero coefficients are listed with their prime factorizations.