A Shintani lift for rigid cocycles

TL;DR

This paper introduces a novel Shintani lift for rigid analytic cocycles of higher weight, linking them to modular forms of half-integral weight and revealing new structural insights and analogies with classical lifts.

Contribution

It constructs a Shintani lift for rigid cocycles, providing explicit Fourier coefficient formulas and connecting rigid cocycles to half-integral weight modular forms.

Findings

Fourier coefficients expressed via residues of cocycles

Similarity between new lift and classical Shintani lift

Strengthens analogy between rigid cocycles and modular forms

Abstract

We construct a Shintani lift for rigid analytic cocycles of higher weight, attaching modular forms of half-integral weight to such cocycles. The expression for the Fourier coefficients of the modular form $\mathcal{RS}(J)$ attached to a cocycle $J$ is given in terms of the residues of $J$, and shares a striking similarity with the expression for the coefficients of the classical Shintani lift $\mathcal{S}(f)$ of an integral weight modular form $f$. This work aligns with the ideas of the nascent $p$-adic Kudla program and strengthens the analogy between rigid cocycles and modular forms.