Weight-Shifting Resolvent Kernels on the Modular Surface

Abstract

The local hypergeometric resolvent kernels and their mixed K-type shifts are due to Fay. On the modular surface, we determine the normalization relating these kernels to the Maass-shifted automorphic resolvent family and equip that family with two compatible extensions: the hyperbolic circular finite part at the diagonal and a parameter-dependent Mellin subtraction matched to the cusp zero mode. On moderate-growth eigenfunctions the resulting meromorphic transform satisfies an explicit spectral-reduction formula. Suppose that the multiplier is trivial at the cusp and that such an eigenfunction has spectral parameter s0 not equal to 1/2. If s0 is a simple zero of the diagonal source coefficient, the continued kernel and Eisenstein series are regular at s0, and the specialized integral converges, then specializing the kernel before integration need not agree with continuing the parameter-dependent finite-part family. Their difference is the relevant constant-term coefficient of the input multiplied by the first spectral-parameter derivative of the kernel's constant Fourier coefficient. For the trivial multiplier, even k greater than 2, raising shifts from k to k+2r, and asymptotic normalization (k-1)/(4 pi), the holomorphic point is s=k/2: specialized integration depends only on the cuspidal part of a modular form, whereas continued finite-part integration acts on the full form. In the case k=12 and t=14, the specialized kernel annihilates the product of y to the sixth power with the holomorphic Eisenstein series of weight 12, while the continued value is minus one twelfth of its Maass raising from weight 12 to weight 14. We also record the Maass shift of the spectral projection and show that the value and first spectral derivative together recover the Maass shift of the reduced resolvent.