[Paper Review] A converse theorem for Borcherds products on $X_0(N)$

Bruinier, Jan Hendrik, Schwagenscheidt, Markus|arXiv (Cornell University)|Jun 25, 2018

Analytic Number Theory Research5 citations

TL;DR

This paper establishes a weak converse theorem for generalized Borcherds products on X₀(N), proving that any Fricke-invariant meromorphic modular form for Γ₀(N) with divisor supported on Heegner divisors and cusps defined over ℚ arises as a Borcherds product of a harmonic Maass form of weight 1/2. It further provides a criterion for the finiteness of the multiplier system in terms of vanishing central L-values of weight 2 newforms, linking arithmetic properties of modular forms to the algebraicity of Fourier coefficients.

ABSTRACT

We show that every Fricke invariant meromorphic modular form for $\Gamma_0(N)$ whose divisor on $X_0(N)$ is defined over $\mathbb{Q}$ and supported on Heegner divisors and the cusps is a generalized Borcherds product associated to a harmonic Maass form of weight $1/2$. Further, we derive a criterion for the finiteness of the multiplier systems of generalized Borcherds products in terms of the vanishing of the central derivatives of $L$-function of certain weight $2$ newforms. We also prove similar results for twisted Borcherds products.

Motivation & Objective

• To establish a converse theorem for generalized Borcherds products on modular curves X₀(N), showing that meromorphic modular forms with divisor supported on Heegner divisors and rational cuspidal divisor are Borcherds products of harmonic Maass forms.
• To characterize when the multiplier system of such a Borcherds product is of finite order, linking this to arithmetic properties of associated modular forms.
• To extend the converse theorem to twisted Borcherds products and derive criteria for algebraicity of Fourier coefficients in terms of Hecke eigenforms and L-functions.
• To prove that the finiteness of the multiplier system is equivalent to the rationality of certain coefficients in the holomorphic part of the harmonic Maass form, using transcendence results on periods of differentials.

Proposed method

• Use the singular theta lift to associate harmonic Maass forms of weight 1/2 for the Weil representation of the lattice Lₙ = ℤ(N)⊕U to meromorphic modular forms on X₀(N).
• Construct the generalized Borcherds product from the principal part of a harmonic Maass form f ∈ H⁺₁/₂,ρₙ, ensuring the divisor matches the given linear combination of Heegner divisors and cusps.
• Apply the ξ-operator to map harmonic Maass forms to cusp forms, and use the fact that the image under ξ₁/₂ determines the multiplier system of the Borcherds product.
• Use orthogonal decomposition of cusp forms and Hecke eigenform theory to decompose the image of ξ₁/₂ and reduce the problem to studying newforms of weight 3/2.
• Leverage the Shimura correspondence to relate weight 3/2 cusp forms to weight 2 newforms, and use the central L-value of the associated L-function to determine whether the multiplier system is finite.
• Apply algebraicity results from transcendence theory (Waldschmidt, Wüstholz, Scholl) to show that the finiteness of the multiplier system is equivalent to the rationality of Fourier coefficients in the holomorphic part of f.

Experimental results

Research questions

• RQ1 Under what conditions is a meromorphic modular form for Γ₀(N) with divisor supported on Heegner divisors and rational cuspidal divisor a generalized Borcherds product of a harmonic Maass form of weight 1/2?
• RQ2 When is the multiplier system of a generalized Borcherds product of finite order, and how can this be characterized arithmetically?
• RQ3 What is the precise relationship between the algebraicity of Fourier coefficients of the holomorphic part of a harmonic Maass form and the vanishing of central L-values of associated weight 2 newforms?
• RQ4 How do twisted Borcherds products behave in this converse setting, and what conditions ensure their algebraicity or finiteness of multiplier systems?

Key findings

• Every Fricke-invariant meromorphic modular form for Γ₀(N) whose divisor on X₀(N) is a linear combination of Heegner divisors and whose cuspidal divisor is defined over ℚ arises as a generalized Borcherds product of a unique harmonic Maass form f ∈ H⁺₁/₂,ρₙ.
• The multiplier system of the generalized Borcherds product is of finite order if and only if the Fourier coefficients a⁺_f(n², n) are algebraic for all n ∈ ℤ.
• The finiteness of the multiplier system is equivalent to the vanishing of the central derivative L′(G, 1) = 0 for the weight 2 newform G associated to the image of f under the ξ-operator via the Shimura correspondence.
• If the image of f under ξ₁/₂ is a simultaneous Hecke eigenform, then the algebraicity of the coefficients a⁺_f(n², n) implies that the corresponding weight 2 newform G has L′(G, 1) = 0.
• The principal part of the harmonic Maass form f can be chosen to have algebraic coefficients if and only if the associated Borcherds product has a multiplier system of finite order.
• The construction of f is achieved by combining Hecke eigenforms and orthogonal decomposition techniques, ensuring that the resulting f has algebraic principal part and maps to the desired cusp form under ξ₁/₂.