On the bad reduction of Hirzebruch-Zagier curvese
(Un exposé de Andreas Langer au séminaire de théorie
des nombres de Chevaleret le 9 décembre 2002)
Résumé :
We consider the Tate-conjecture for divisors in finite
characteristic for a certain class of Hilbert modular surfaces
where
all Hilbert modular forms are lifts of elliptic modular forms.
If p splits in the underlying real quadratic field the Conjecture
is shown by considering components in the reduction of a non-compact
Hirzebruch-Zagier curve(the image of a modular curve).
If p is inert the whole cusidal cohomology is supposed to be covered
by cycle classes in char. p. We construct-for each isotypic component
under the action of the Hecke-algebra- a new cycle class by considering
a Shimura curve having bad reduction at p, the components in the reduction
ly in the supersingular locus of the surface and give rise to new cycle
classes