Arithmetic Gan-Gross-Prasad conjecture for RSZ unitary Shimura curves

TL;DR

This paper proves a case of the arithmetic Gan-Gross-Prasad conjecture for RSZ unitary Shimura curves by reinterpreting Xue's results and connecting them with the Gross-Zagier formula, advancing understanding of special cycles and automorphic forms.

Contribution

It reinterprets Xue's results in terms of RSZ Shimura varieties and proves a case of the variant conjecture, linking it to the Gross-Zagier formula.

Findings

Established a case of the variant Gan-Gross-Prasad conjecture for RSZ Shimura curves.

Connected the variant conjecture with the Gross-Zagier formula.

Reinterpreted Xue's results within the framework of RSZ Shimura varieties.

Abstract

Xue proved an equational refinement of the unitary Shimura curve case of the arithmetic Gan-Gross-Prasad conjecture via the Gross-Zagier formula for quaternionic Shimura curves. On the other hand, Rapoport, Smithling and Zhang posed a variant of the conjecture, using modified PEL type Shimura varieties, which we call RSZ Shimura varieties. We reinterpret the result of Xue in terms of the modified Shimura curves. We then use the reinterpretation to prove a case of the variant of the conjecture. Our result combined with the work of Xue establishes a connection between the variant and the Gross-Zagier formula.