Tate classes on a product of two Picard modular surfaces (Q1876235)

The author computes the space of codimension 2 Tate classes on a product of two Picard modular surfaces in terms of automorphic representations on \(\text{GL}(n)\) for \(n < 4\). To each cycle \(Z\) on a smooth projective variety \(X\), defined over a number field \(E\), of dimension \(i\) defined over an extension field \(L\), there is associated a cohomology class by Poincaré duality. The author studies the Tate class by identifying Tate classes on \(X\). The work of Murty and Prasad on the same problem for a product of Hilbert modular surfaces is applied here to the case of two modular surfaces relative to two distinct real quadratic fields. When the representations are not automorphically induced, the corresponding Tate classes are shown to be abelian.