| Résume | We consider the Tate-conjecture for divisors in finitecharacteristic for a certain class of Hilbert modular surfaceswhereall Hilbert modular forms are lifts of elliptic modular forms.If p splits in the underlying real quadratic field the Conjectureis shown by considering components in the reduction of a non-compactHirzebruch-Zagier curve(the image of a modular curve).If p is inert the whole cusidal cohomology is supposed to be coveredby cycle classes in char. p. We construct-for each isotypic componentunder the action of the Hecke-algebra- a new cycle class by consideringa Shimura curve having bad reduction at p, the components in the reductionly in the supersingular locus of the surface and give rise to new cycleclasses |
