Arithmetic of K3 surface
نویسنده
چکیده
We review recent developments in the arithmetic of K3 surfaces. Our focus lies on aspects of modularity, Picard number and rational points. Throughout we emphasise connections to geometry.
منابع مشابه
Arithmetic of a singular K3 surface
This paper is concerned with the arithmetic of the elliptic K3 surface with configuration [1,1,1,12,3*]. We determine the newforms and zeta-functions associated to X and its twists. We verify conjectures of Tate and Shioda for the reductions of X at 2 and 3.
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A K3 surface X is a compact complex surface with KX ∼ 0 and H (X,OX) = 0. An Enriques surface is a compact complex surface with H(Y,OY ) = H (Y,OY ) = 0 and 2KY ∼ 0. The universal covering of an Enriques surface is a K3 surface. Conversely every quotient of a K3 surface by a free involution is an Enriques surface. Here a free involution is an automorphism of order 2 without any fixed points. Th...
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Arithmetic of K3 surfaces defined over finite fields is investigated. In particular, we show that any K3 surface X of finite height over a finite field k of characteristic p ≥ 5 has a quasi-canonical lifting Z to characteristic 0, and that for any such Z, the endormorphism algebra of the transcendental cycles V (Z), as a Hodge module, is a CM field over Q. The Tate conjecture for the product of...
متن کاملArithmetic of K3 surfaces
We review recent developments in the arithmetic of K3 surfaces. Our focus lies on aspects of modularity, Picard number and rational points. Throughout we emphasise connections to geometry.
متن کاملar X iv : 0 70 9 . 19 79 v 1 [ m at h . A G ] 1 3 Se p 20 07 K 3 Surfaces of Finite Height over Finite Fields ∗ †
Arithmetic of K3 surfaces defined over finite fields are investigated. In particular, we show that any K3 surface X of finite height over a finite field k of characteristic p ≥ 5 has a quasi-canonical lifting Z to characteristic 0, and that the endormorphism algebra of the transcendental cycles V (Z), as a Hodge module, is a CM field over Q. We also prove the Tate conjecture for any powers of s...
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