Tate classes and \(L\)-functions on a product of a quaternionic Shimura surface and a Picard modular surface
The author determines the space of Tate classes on the product of a quaternionic Shimura surface and a Picard modular surface in terms of automorphic representations. More precisely, let \(S\) be the quasi-projective Shimura surface associated with a quaternion algebra \(D\), split at exactly \(2\) infinite places, over a totally real field \(F\), and some level structure. Let \(\mathbf S\) be the compactified Picard modular surface associated with the quasi-split unitary similitude group over \(\mathbb Q\) associated with a quadratic imaginary extension \(L\) of \(\mathbb Q\) and a Hermitian inner product on \(L^3\) of signature \((2,1)\). The author describes the space of Tate classes of the product \(S_M\times \mathbf S_M\), where \(M\) is the composite of the reflex field of \(S\) and of \(L\), in terms of the decomposition, resulting from the Hecke algebra operation, of the relevant cohomology group into automorphic representations. The fields of definition of the Tate classes are determined. In special cases, it is proved that the dimension of the space of Tate classes is given by the order at \(s=3\) of the corresponding \(L\)-function, as predicted by the Tate conjecture.
scientific article; zbMATH DE number 5285723
scientific article