Unconditional construction of K3 surfaces over finite fields with given L-function in large characteristic

Existence of supersingular reduction for families of $K3$ surfaces with large Picard number in positive characteristic

“…Assuming the existence of potential semistable reduction, Taelman conditionally proved the existence of K3 surfaces over finite fields with given L-function, up to finite extensions of the base field [28]. When the characteristic of the base field is not too small, the author recently proved that Taelman's results hold unconditionally [10]. As an application, the author proved the following theorem.…”

“…In Section 4, using deformation theory, we show Theorem 1.3 comes down to show the existence of K3 surfaces X over F p satisfying ρ(X) = 22 − 2h(X) and h(X) = h. Assuming semistable reduction, Taelman conditionally proved the existence of K3 surfaces over finite fields with given L-function, up to finite extensions of the base field; see [28]. When the characteristic of the base field is not too small, the author recently proved that Taelman's results hold unconditionally; see [10]. Using these results, in Section 5, we construct K3 surfaces X over F p satisfying ρ(X) = 22 − 2h(X) and h(X) = h for large p and each 2 ≤ h ≤ 10; see Proposition 5.2.…”

Degeneration of Kummer surfaces

“…This applies in particular to Kummer surfaces, at least after a finite extension of the ground field. Then there exists a finite extension L/K and a proper algebraic space Y → Spec O K which is a strict Kulikov model of X L (see [19], Proposition 3.1 together with [14], Propositions 2.3 and 2.4). In general, these algebraic spaces cannot be guaranteed to be schemes (see [19], proof of Proposition 3.1), so this does not imply potential semistable reduction for Kummer surfaces in the category of schemes.…”

“…Furthermore, our method also applies in the case p = 3. The author is grateful to Professor C. Liedtke for bringing the paper [14] to his attention.…”

Zeta functions of nondegenerate hypersurfaces in toric varieties via controlled reduction in p-adic cohomology

“…In dimension 2, they are K3 surfaces, whose zeta functions are of computational interest for various reasons. For instance, these zeta functions can (potentially) be used to establish the infinitude of rational curves on a K3 surface (see the introduction to [CT14] for discussion); there has also been recent work on analogues of the Honda-Tate theorem, establishing conditions under which particular zeta functions are realized by K3 surfaces [Tae16,Ito16].…”