Tate conjecture for twisted Picard modular surfaces (Q531839)

This paper starts with a clear and brief statement of the Tate conjecture on algebraic cycles. Now let \(F\) be a totally real number field. Let \(E\) be a quadratic imaginary extension of \(F\). Let \(S_K= S_{K,G}\) be the Picard modular surface associated to a unitary similitude group \(G = \text{GU}(3)\) defined relative to \(E/F\) and a compact subgroup \(K\) of \(G(A_{F,f})\), where \(A_{F,f}\) is the ring of finite adeles of \(F\). The paper considers a twist \(S'_K\) of \(S_K\), and proves that the first part of the Tate conjecture for \(S'_K\) (the Tate classes of \(S'_K\) are algebraic) is true since it is known for \(S_K\). Further, results on the second part of the Tate conjecture (meromorphic continuation of the relevant \(L\)-function, functional equation, order of pole) are obtained in some cases for the same twist \(S'_K\).