Special cycles on unitary Shimura varieties II: Global theory

Abstract

Abstract We introduce moduli spaces of abelian varieties which are arithmetic models of Shimura varieties attached to unitary groups of signature ( n - 1 , 1 ) $(n-1, 1)$ . We define arithmetic cycles on these models and study their intersection behavior. In particular, in the non-degenerate case, we prove a relation between their intersection numbers and Fourier coefficients of the derivative at s = 0 of a certain incoherent Eisenstein series for the group U ( n , n ) $\textup {U}(n, n)$ . This is done by relating the arithmetic cycles to their formal counterpart from part I [Invent. Math. 184 (2011), 629–682] via non-archimedean uniformization, and by relating the Fourier coefficients to the derivatives of representation densities of hermitian forms. The result then follows from the main theorem of part I and a counting argument.