Weight-Shifting Operators of Hypergeometric Type for Maass Forms

TL;DR

This paper develops hypergeometric type weight-shifting integral operators for Maass forms on SL(2,Z), using automorphic kernels derived from solutions to hypergeometric differential equations, with implications for automorphic representation theory.

Contribution

It introduces a new class of weight-shifting operators for Maass forms based on hypergeometric functions, linking automorphic kernels to hypergeometric differential equations.

Findings

Operators are constructed via hypergeometric solutions to a Papperitz-Riemann equation.

Convergence conditions are established based on asymptotic analysis of hypergeometric solutions.

The operators potentially relate to Hecke algebra actions and motivic structures.

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.