A $p$-adic interpolation of the Cogdell lift

TL;DR

This paper develops a $p$-adic interpolation framework for the Cogdell lift, connecting special cycles on Picard modular surfaces to elliptic modular forms, and constructs related $p$-adic cohomology classes.

Contribution

It introduces a new $p$-adic interpolation of the Cogdell lift, extending it to higher weights and levels, and constructs associated $p$-adic cohomology classes.

Findings

$p$-adic interpolation of the Cogdell lift achieved.

Higher weight cycles in Kuga-Sato varieties constructed and shown to be modular.

A Hida family interpolating Cogdell lifts in weight and level established.

Abstract

In this paper we obtain several results related to the $p$-adic interpolation of the classical Cogdell lift, mapping special cycles on Picard modular surfaces to elliptic modular forms. The results have a three-fold nature: in the first part of the paper, we $p$-adically interpolate the adjoint Kudla lift, exploiting the previously constructed $\Lambda$-adic Kudla lift. In the second part, we construct higher weight cycles in Kuga-Sato varieties attached to Picard modular surfaces, and show modularity of the generating series of these cycles, thus obtaining a higher weight analogue of the Cogdell lift. Finally, we apply the formalism introduced by Loeffler to construct $p$-adic analytic cohomology classes of special cycles, whose generating series is proved to be a Hida family interpolating the Cogdell lifts in the weight and level variables.