A Shimura-Shintani correspondence for rigid analytic cocycles of higher weight

TL;DR

This paper initiates a systematic study of higher weight additive rigid meromorphic cocycles, extending previous work on weight two, and constructs a kernel function linking modular forms of half-integer weight to these cocycles.

Contribution

It classifies certain higher weight rigid meromorphic cocycles and constructs an explicit kernel function establishing a Shimura-Shintani correspondence.

Findings

Classification of weight 2k rigid meromorphic cocycles

Explicit kernel function for Shimura-Shintani correspondence

Extension of known correspondences to higher weights

Abstract

This paper takes the first steps towards a systematic study of additive rigid meromorphic cocycles of higher weight. These were introduced by Darmon and Vonk, who focused on multiplicative and weight two cocycles. After classifying certain rigid meromorphic cocycles of weight $2k$, we construct an explicit holomorphic kernel function realising a Shimura-Shintani style correspondence from modular forms of weight $k+1/2$ and level $4p^2$ to rigid analytic cocycles of weight $2k$ on SL$_2(\mathbb{Z}[1/p])$.