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Harmonic Maass forms associated with CM newforms

Stephan Ehlen, Yingkun Li, Markus Schwagenscheidt

TL;DR

The paper tackles the algebraicity question for holomorphic Fourier coefficients of harmonic Maass forms mapped from CM newforms via ξ-operators. It develops a regularized theta-lift framework that yields ξ-preimages whose holomorphic parts have coefficients in the CM field and transform compatibly under the Galois action. By analyzing CM cusp forms, binary theta series, and Borcherds–Shimura lifts, the authors prove that these coefficients are algebraic and provide a concrete, Galois-equivariant construction of good preimages; they also give a numerical example illustrating the method and connecting to existing results. The work advances understanding of when harmonic Maass forms have algebraic holomorphic parts and clarifies the arithmetic nature of their CM-associated coefficients, with explicit normalizations via Petersson norms and CM-period relations.

Abstract

In this paper, we use a regularized theta lifting to construct harmonic Maass forms corresponding to binary theta functions of weight $k \ge 2$ under the $ξ$-operator. As a result, we show that their holomorphic parts have algebraic Fourier coefficients, with compatible Galois action. As an application, we prove rationality properties of coefficients of harmonic Maaass forms corresponding to CM newforms, answering a question of Bruinier, Ono and Rhoades.

Harmonic Maass forms associated with CM newforms

TL;DR

The paper tackles the algebraicity question for holomorphic Fourier coefficients of harmonic Maass forms mapped from CM newforms via ξ-operators. It develops a regularized theta-lift framework that yields ξ-preimages whose holomorphic parts have coefficients in the CM field and transform compatibly under the Galois action. By analyzing CM cusp forms, binary theta series, and Borcherds–Shimura lifts, the authors prove that these coefficients are algebraic and provide a concrete, Galois-equivariant construction of good preimages; they also give a numerical example illustrating the method and connecting to existing results. The work advances understanding of when harmonic Maass forms have algebraic holomorphic parts and clarifies the arithmetic nature of their CM-associated coefficients, with explicit normalizations via Petersson norms and CM-period relations.

Abstract

In this paper, we use a regularized theta lifting to construct harmonic Maass forms corresponding to binary theta functions of weight under the -operator. As a result, we show that their holomorphic parts have algebraic Fourier coefficients, with compatible Galois action. As an application, we prove rationality properties of coefficients of harmonic Maaass forms corresponding to CM newforms, answering a question of Bruinier, Ono and Rhoades.
Paper Structure (13 sections, 12 theorems, 148 equations)

This paper contains 13 sections, 12 theorems, 148 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Modular forms for the Weil representation
  4. CM cusp forms
  5. Binary theta series
  6. Quadratic spaces and theta functions
  7. Main Results
  8. Algebraicity of regularized inner products.
  9. Comparing Petersson norms
  10. Preimages of CM newforms
  11. A numerical example
  12. Petersson inner products as special values of theta lifts
  13. Evaluation of theta lifts

Key Result

Theorem 1.1

For every Hecke character $\varrho$ of weight $k \ge 2$, there exists $\tilde{\vartheta}_\varrho \in H_{2-k}(N,\overline{\chi})$ good for $\vartheta_\varrho$ such that its holomorphic part $\tilde{\vartheta}_\varrho^+$ satisfies for all $\sigma \in \mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$. Here $\varrho^\sigma$ is the Hecke character defined in eq:chisigc.

Theorems & Definitions (30)

  • Theorem 1.1
  • Corollary 1.2
  • Remark 1.3
  • Remark 1.4
  • Conjecture 1.5
  • Lemma 2.1
  • Proposition 2.2
  • Lemma 2.3
  • proof
  • Remark 2.4
  • ...and 20 more