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Weight-Shifting Operators of Hypergeometric Type for Maass Forms

Seung Ju Lee

TL;DR

The paper develops a family of weight-shifting integral operators that map weight k Maass forms to weight t forms for Γ = SL(2,Z) by periodizing a seed kernel built from a covariant factor and a G-invariant kernel. The key theoretical advance is the reduction of the weight-t Laplacian eigenproblem for the kernel to a Papperitz–Riemann equation, which, via a transformation to Gauss hypergeometric form, reveals the operator as hypergeometric-type. A careful analysis of asymptotics, singularities, cusp behavior, and regularization shows how to choose the subdominant hypergeometric solution to ensure absolute convergence of the Poincaré series and to construct a well-defined automorphic kernel. The work also discusses intertwining properties, the role of the spectral parameter, and potential links to Hecke algebras and motivic interpretation, highlighting the arithmetic-geometric significance of hypergeometric structures in automorphic contexts.

Abstract

This paper constructs weight-shifting integral operators for Maass forms on the full modular group SL(2,Z). Under the weight parity condition t = k (mod 2), the operator utilizes an automorphic kernel constructed via Poincare series from a seed kernel. The seed kernel is defined as the product of a covariant factor and an invariant factor with respect to the diagonal action of SL(2,R). A spectral condition is imposed that the kernel must be an eigenfunction of the weight-t hyperbolic Laplacian. This problem reduces to an ordinary differential equation (ODE) for the invariant factor, which is identified as a Papperitz-Riemann equation. By transforming this equation into the Gauss Hypergeometric Differential Equation (HDE), the hypergeometric type of the operator is established. Analysis of the asymptotic behavior of the hypergeometric solutions yields the convergence conditions for the automorphic kernel and the integral operator, requiring the selection of the subdominant solution. The operator is defined via regularization based on the Hyperbolic Cauchy Principal Value to handle the diagonal singularities of the kernel. The automorphy of the transformed function is verified under the assumption of absolute convergence. A necessary condition for the intertwining property, namely the coincidence of the kernel's spectral parameter and the input Maass form's Laplacian eigenvalue, is conditionally derived. Analysis of the smoothness and the behavior at the cusp of the transformed function is deferred to subsequent research. Finally, the relevance of the hypergeometric structure to Hecke algebra compatibility and motivic interpretation is discussed.

Weight-Shifting Operators of Hypergeometric Type for Maass Forms

TL;DR

The paper develops a family of weight-shifting integral operators that map weight k Maass forms to weight t forms for Γ = SL(2,Z) by periodizing a seed kernel built from a covariant factor and a G-invariant kernel. The key theoretical advance is the reduction of the weight-t Laplacian eigenproblem for the kernel to a Papperitz–Riemann equation, which, via a transformation to Gauss hypergeometric form, reveals the operator as hypergeometric-type. A careful analysis of asymptotics, singularities, cusp behavior, and regularization shows how to choose the subdominant hypergeometric solution to ensure absolute convergence of the Poincaré series and to construct a well-defined automorphic kernel. The work also discusses intertwining properties, the role of the spectral parameter, and potential links to Hecke algebras and motivic interpretation, highlighting the arithmetic-geometric significance of hypergeometric structures in automorphic contexts.

Abstract

This paper constructs weight-shifting integral operators for Maass forms on the full modular group SL(2,Z). Under the weight parity condition t = k (mod 2), the operator utilizes an automorphic kernel constructed via Poincare series from a seed kernel. The seed kernel is defined as the product of a covariant factor and an invariant factor with respect to the diagonal action of SL(2,R). A spectral condition is imposed that the kernel must be an eigenfunction of the weight-t hyperbolic Laplacian. This problem reduces to an ordinary differential equation (ODE) for the invariant factor, which is identified as a Papperitz-Riemann equation. By transforming this equation into the Gauss Hypergeometric Differential Equation (HDE), the hypergeometric type of the operator is established. Analysis of the asymptotic behavior of the hypergeometric solutions yields the convergence conditions for the automorphic kernel and the integral operator, requiring the selection of the subdominant solution. The operator is defined via regularization based on the Hyperbolic Cauchy Principal Value to handle the diagonal singularities of the kernel. The automorphy of the transformed function is verified under the assumption of absolute convergence. A necessary condition for the intertwining property, namely the coincidence of the kernel's spectral parameter and the input Maass form's Laplacian eigenvalue, is conditionally derived. Analysis of the smoothness and the behavior at the cusp of the transformed function is deferred to subsequent research. Finally, the relevance of the hypergeometric structure to Hecke algebra compatibility and motivic interpretation is discussed.

Paper Structure

This paper contains 52 sections, 63 theorems, 371 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and Definitions
  3. Construction of the Seed Kernel
  4. The Covariant Factor $P$
  5. 1. Transformation of the $q$-dependent part ($P_q$)
  6. 2. Transformation of the Weight-dependent part ($P_W$)
  7. 3. Calculation of $TF_W$
  8. 4. Conclusion
  9. Condition on Weight Parity
  10. The Spectral Eigenvalue Problem and its Solution
  11. Reduction of the Laplacian Operator
  12. Properties of the Reduced Operator $\Delta_Q$
  13. Reduction to an Ordinary Differential Equation
  14. Solution via the Hypergeometric Differential Equation
  15. The singular point at $z=0$
  16. ...and 37 more sections

Key Result

Lemma 3.1

For any $g \in \Gamma$, $q \in \mathbb{C}$, and $\tau_1, \tau_2 \in \mathbb{H}$, the factor $P(\tau_1, \tau_2)$ transforms as follows: where $j_k(g,\tau)$ is the weight-$k$ automorphy factor, and the total phase factor $\mathcal{E}_{\text{total}}$ is:

Theorems & Definitions (150)

  • Definition 2.1: The Full Modular Group
  • Definition 2.2: Fundamental Domain for $SL(2,\mathbb{Z})$
  • Definition 2.3: Weight-$k$ Hyperbolic Laplacian
  • Definition 2.4: Space of Maass Forms of Weight $k$
  • Definition 3.1: Phase Factors from Principal Branch Conventions
  • Definition 3.2: The Covariant Factor P
  • Remark 3.1: Computational Form of P
  • Lemma 3.1: Phase-Adjusted Covariance of the Factor $P$
  • proof
  • Definition 3.3: The Seed Kernel $K_{k,t,q}$
  • ...and 140 more