1,509 research outputs found
The Tate conjecture for K3 surfaces over finite fields
Artin's conjecture states that supersingular K3 surfaces over finite fields
have Picard number 22. In this paper, we prove Artin's conjecture over fields
of characteristic p>3. This implies Tate's conjecture for K3 surfaces over
finite fields of characteristic p>3. Our results also yield the Tate conjecture
for divisors on certain holomorphic symplectic varieties over finite fields,
with some restrictions on the characteristic. As a consequence, we prove the
Tate conjecture for cycles of codimension 2 on cubic fourfolds over finite
fields of characteristic p>3.Comment: 20 pages, minor changes. Theorem 4 is stated in greater generality,
but proofs don't change. Comments still welcom
The Tate conjecture for K3 surfaces in odd characteristic
We show that the classical Kuga-Satake construction gives rise, away from
characteristic 2, to an open immersion from the moduli of primitively polarized
K3 surfaces (of any fixed degree) to a certain regular integral model for a
Shimura variety of orthogonal type. This allows us to attach to every polarized
K3 surface in odd characteristic an abelian variety such that divisors on the
surface can be identified with certain endomorphisms of the attached abelian
variety. In turn, this reduces the Tate conjecture for K3 surfaces over
finitely generated fields of odd characteristic to a version of the Tate
conjecture for certain endomorphisms on the attached Kuga-Satake abelian
variety, which we prove. As a by-product of our methods, we also show that the
moduli stack of primitively polarized K3 surfaces of degree 2d is
quasi-projective and, when d is not divisible by p^2, is geometrically
irreducible in characteristic p. We indicate how the same method applies to
prove the Tate conjecture for co-dimension 2 cycles on cubic fourfolds
Supersingular K3 Surfaces are Unirational
We show that supersingular K3 surfaces in characteristic are related
by purely inseparable isogenies. This implies that they are unirational, which
proves conjectures of Artin, Rudakov, Shafarevich, and Shioda. As a byproduct,
we exhibit the moduli space of rigidified K3 crystals as an iterated
-bundle over . To complete the picture, we also
establish Shioda-Inose type isogeny theorems for K3 surfaces with Picard rank
in positive characteristic.Comment: 31 pages; many details added, final versio
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