Unitary cycles on Shimura curves and the Shimura lift II

TL;DR

This paper links the Shimura lift of certain modular forms to generating series of unitary divisors on Shimura curves, combining geometric and arithmetic methods to produce new modular generating series with coefficients in arithmetic Chow groups.

Contribution

It identifies the Shimura lift with a generating series of unitary divisors, extending previous work and providing new examples of modular series in arithmetic Chow groups.

Findings

Shimura lift corresponds to a generating series of unitary divisors.

New modular generating series with coefficients in arithmetic Chow groups.

Combined geometric and archimedian analysis to establish the main theorem.

Abstract

We consider two families of arithmetic divisors defined on integral models of Shimura curves. The first was studied by Kudla, Rapoport and Yang, who proved that if one assembles these divisors in a formal generating series, one obtains the q-expansion of a modular form of weight 3/2. The present work concerns the Shimura lift of this modular form: we identify the Shimura lift with a generating series comprised of unitary divisors, which arose in recent work of Kudla and Rapoport regarding cycles on Shimura varieties of unitary type. In the prequel to this paper, the author considered the geometry of the two families of cycles; these results are combined with the archimedian calculations found in this work in order to establish the theorem. In particular, we obtain new examples of modular generating series whose coefficients lie in arithmetic Chow groups of Shimura varieties.