Period relations and the Tate conjecture for Hilbert modular surfaces
Harder, Langlands and Rapoport had proved the Tate conjecture for algebraic cycles for the non-CM-submotives of \(H^ 2\) of Hilbert modular surfaces [\textit{G. Harder}, \textit{R. P. Langlands}, \textit{M. Rapoport}, J. Reine Angew. Math. 366, 53--120 (1986; Zbl 0575.14004)]. The authors consider Hilbert modular forms of CM-type and prove the Tate conjecture for the CM-submotives in \(H^ 2\). This gives the full Tate conjecture asserting the algebraicity of all Tate cycles on Hilbert modular surfaces over an arbitrary number field. The authors also relate, in the CM case, the number of independent divisor classes to the order of pole of the \(L\)-function of the surface at the edge of convergence. The results have interesting corollaries about the Picard group of the surface. There exist non-trivial divisors defined over metabelian fields which are not in the linear span of the Hirzebruch-Zagier cycles, canonical divisors and the cycles coming from the desingularization of the surface. The methods are quite different from the methods of Harder, Langlands and Rapoport.
scientific article; zbMATH DE number 4025577
scientific article