Unitary cycles on Shimura curves and the Shimura lift I

TL;DR

This paper explores the relationship between orthogonal and unitary special cycles on Shimura curves, demonstrating that they are connected through a formal Shimura lift, thus linking different types of cycles via modular form correspondences.

Contribution

It establishes a formal relationship between orthogonal and unitary special cycles on Shimura curves using the Shimura lift, extending previous work on modular forms and Shimura varieties.

Findings

Orthogonal and unitary cycles are related by the Shimura lift.

The work extends Kudla-Rapoport's studies to new cycle families.

A formal version of the Shimura lift connects these cycles.

Abstract

This paper concerns two families of divisors, which we call the `orthogonal' and `unitary' special cycles, defined on integral models of Shimura curves. The orthogonal family was studied extensively by Kudla-Rapoport-Yang, who showed that they are closely related to the Fourier coefficients of modular forms of weight 3/2, while the `unitary' divisors are analogues of cycles appearing in more recent work of Kudla-Rapoport on unitary Shimura varieties. Our main result shows that these two families are related by (a formal version of) the Shimura lift.