Hecke and Galois Properties of Special Cycles on Unitary Shimura Varieties

Abstract

We define and study a collection of special cycles on certain non-PEL Shimura varieties for $U(2,1) \times U(1,1)$ that appear naturally in the context of the recent conjectures of Gan, Gross and Prasad on restrictions of automorphic forms for unitary groups and conjectural generalizations of the Gross--Zagier formula. We prove a combinatorial formula expressing the Galois action in terms of the distance function on the Bruhat--Tits buildings for these groups. In addition, we calculate explicitly the Hecke polynomial appearing in the congruence relation conjectured by Blasius and Rogawski. Using methods and recent results of Koskivirta, we prove the congruence relation in this case. Finally, using the action of the local Hecke algebra on the Bruhat--Tits building, we establish explicit relations (distribution relations) between the Hecke action and the Galois action on the special cycles. These relations are key for a work in progress by the author on constructing a new Euler system from these cycles and using it to prove instances of the Bloch{Kato{ Beilinson conjecture for certain Galois representations associated to automorphic forms on unitary groups.