[Paper Review] Traces of CM values of modular functions and related topics

Jan Hendrik Bruinier|ArXiv.org|Oct 14, 2005

Holomorphic and Operator Theory27 references3 citations

TL;DR

This paper generalizes Zagier's theory of traces of CM values of modular functions to modular curves of arbitrary genus using a theta lift construction. It establishes that the generating series of these traces is a meromorphic modular form of weight 3/2, and proves an asymptotic formula showing that the trace is closely approximated by a sum over reduced quadratic forms, with the error term converging to -24 in the limit as the discriminant grows.

ABSTRACT

The purpose of this note is to report on recent joint work with J. Funke, P. Jenkins, and K. Ono on the traces of CM values of modular functions and some applications.

Motivation & Objective

• To extend Zagier's results on traces of singular moduli from genus zero to modular curves of arbitrary genus.
• To establish a theta lift construction linking modular functions on Γ(1) to weight 3/2 modular forms on Γ₀(4).
• To derive exact and asymptotic formulas for the traces of CM values of modular functions.
• To analyze the arithmetic and geometric significance of the generating series of these traces.
• To connect the trace formulas to spectral theory and the equidistribution of CM points via the Atkin functional.

Proposed method

• Utilizes the Kudla-Millson theta kernel to define a theta integral that converges absolutely despite poles in the modular function.
• Applies the theta lift to map weight 0 modular functions (e.g., J(τ) = j(τ) − 744) to weight 3/2 modular forms on Γ₀(4).
• Uses Serre duality and Hecke summation to define Poincaré series in weight 3/2, even when they do not converge classically.
• Applies the Kohnen projection operator to restrict to the plus space, ensuring compatibility with the Shimura lift framework.
• Employs spectral theory and resolvent kernels to analytically continue Poincaré series and compute their Fourier expansions.
• Relies on the equidistribution of CM points and bounds on L-functions to derive asymptotic estimates for the trace.

Experimental results

Research questions

• RQ1 Can the generating series of traces of CM values of modular functions be described as a meromorphic modular form of weight 3/2?
• RQ2 How does the theta lift construction extend Zagier’s original result beyond genus zero modular curves?
• RQ3 What is the asymptotic behavior of the trace of CM values as the discriminant D tends to infinity?
• RQ4 To what extent can the trace be approximated by a sum over reduced quadratic forms?
• RQ5 What is the role of the Atkin functional in regularizing the average value of the modular function over the fundamental domain?

Key findings

• The generating series of the traces of CM values of J(τ) is a meromorphic modular form of weight 3/2 for Γ₀(4), explicitly given by g(τ) = η(τ)²E₄(4τ)/(η(2τ)η(4τ)⁶).
• The trace t_J(D) satisfies the asymptotic formula t_J(D) = (−1)^D e^{π√D} + O(e^{cπ√D}) for any c > 1/2 as D → ∞.
• The trace t_J(D) is asymptotically approximated by a sum over reduced quadratic forms with imaginary part > 1, with the difference converging to −24.
• The limit lim_{D→∞} (1/H(D)) (t_J(D) − ∑_{Q∈Q_D^{red}, Im(α_Q)>1} e(−α_Q)) = −24 holds, where H(D) is the class number.
• This limit arises as the regularized average of J(z) over the fundamental domain, known as the Atkin functional.
• The result is generalized to all weakly holomorphic modular functions f ∈ M₀!(Γ(1)) with appropriate modifications.