[Paper Review] Traces of CM values and cycle integrals of polyharmonic Maass forms

Toshiki Matsusaka|arXiv (Cornell University)|May 5, 2018

Advanced Mathematical Identities20 references3 citations

TL;DR

This paper constructs a basis for polyharmonic weak Maass forms of half-integral weight and derives arithmetic formulas for their Fourier coefficients in terms of traces of CM values and cycle integrals. It generalizes Zagier's work on generating functions of CM traces to polyharmonic Maass forms, unifying results for modular functions like the j-invariant and log|η(z)|⁴ under a common framework via Laurent coefficients of Eisenstein series.

ABSTRACT

Lagarias and Rhoades generalized harmonic Maass forms by considering forms which are annihilated by a number of iterations of the action of the xi-operator. In our previous work, we considered polyharmonic weak Maass forms by allowing the exponential growth at cusps, and constructed a basis of the space of such forms. This paper focuses on the case of half-integral weight. We construct a basis as an analogue of our work, and give arithmetic formulas for the Fourier coefficients in terms of traces of CM values and cycle integrals of polyharmonic weak Maass forms. These results put the known results into a common framework.

Motivation & Objective

• To extend the theory of traces of CM values and cycle integrals to polyharmonic weak Maass forms of half-integral weight.
• To construct a basis for the space of such forms, analogous to prior work on integral weight.
• To derive explicit arithmetic formulas for Fourier coefficients in terms of traces of CM values and cycle integrals.
• To unify known results—such as those for the j-invariant and log|η(z)|⁴—under a single framework using Laurent coefficients of Eisenstein series.

Proposed method

• Define polyharmonic weak Maass forms of half-integral weight as smooth functions on the upper half-plane that are invariant under the action of Γ₀(4), annihilated by r iterations of the ξ-operator, and of polynomial growth at cusps.
• Use the holomorphic projection of Eisenstein series to extract the holomorphic part of polyharmonic Maass forms.
• Compute the Laurent coefficients at s=1 or s=3/4 of the Eisenstein series G_D,m(z,s) to relate them to Fourier coefficients of the holomorphic part.
• Apply the functional equation and spectral decomposition of the Poincaré series to express coefficients in terms of class numbers and L-functions.
• Use the Kronecker symbol and the Dedekind eta function to handle the weight 1/2 setting and define the slash operator accordingly.
• Leverage known expansions of real-analytic Eisenstein series and their behavior near s=1 to extract the required Laurent coefficients.

Experimental results

Research questions

• RQ1 Can the generating function of traces of CM values for polyharmonic Maass forms of half-integral weight be modular, generalizing Zagier’s results for harmonic Maass forms?
• RQ2 How can the Fourier coefficients of polyharmonic weak Maass forms of half-integral weight be expressed arithmetically in terms of CM values and cycle integrals?
• RQ3 What is the role of the Laurent coefficient of Eisenstein series in connecting arithmetic data (class numbers, L-values) to modular forms of half-integral weight?
• RQ4 Can the generating function ∑ₙ Tr_d(f)q⁻ⁿ be realized as the holomorphic part of a polyharmonic Maass form for any modular-invariant function f?
• RQ5 How do the traces Tr_d(f) for f = −log(y|η(z)|⁴) relate to L-values and class numbers in the half-integral weight case?

Key findings

• The Fourier coefficients of the holomorphic part of F_{1/2,0,0}(z) are given by 3/√d times the trace Tr_d,1(1), linking them to class numbers via the Kronecker-Hurwitz formula.
• For fundamental discriminant D ≠ 1, the trace Tr_d,D(1) vanishes if d is not a square, and equals √|dD|L_D(1)/π if d is a square, establishing a direct link to L-functions.
• The generating function ∑_{d<0} Tr_d(−log(y|η(z)|⁴))q⁻ᵈ is realized as the holomorphic part of a polyharmonic Maass form of weight 1/2 and depth 3/2.
• The Laurent coefficient LC_{s=1}^{-1}[G_{D,0}(z,s)]^hol vanishes identically, implying ∑_{d<0} Tr_d,D(1)q⁻ᵈ = 0 for D ≠ 1.
• For f(z) = −log(y|η(z)|⁴), the generating function ∑_{d<0} Tr_d(f)q⁻ᵈ is shown to be a modular form via the holomorphic projection of Eisenstein series.
• The coefficient of q⁻ᵈ in the holomorphic part of F_{1/2,0,0}(z) is proportional to Tr_d,D(−log(y|η(z)|⁴)) for D > 1, with proportionality factor 3/(√(dD)L_D(1)).