In this paper we give a characterization of the height of K3 surfaces in characteristic $p>0$. This enables us to calculate the cycle classes in families of K3 surfaces of the loci where the height is at least $h$. The formulas for such loci can be seen as generalizations of the famous formula of Deuring for the number of supersingular elliptic curves in characteristic $p$. In order to describe the tangent spaces to these loci we study the first cohomology of higher closed forms.

This article confirms a consequence of the non-abelian Iwasawa main conjecture. It is proved that under a technical condition the étale cohomology groups $H^1({𝒪}_K[1/S],H^i(\overline{X},{\mathbb{Q}}_p\left(j\right)))$, where $X\to Spec{𝒪}_K[1/S]$ is a smooth, projective scheme, are generated by twists of norm compatible units in a tower of number fields associated to $H^i(\overline{X},{\mathbb{Z}}_p\left(j\right))$. Using the “Bloch-Kato-conjecture” a similar result is proven for motivic cohomology with finite coefficients.

We consider a short sequence of hermitian vector bundles on some arithmetic variety. Assuming that this sequence is exact on the generic fiber we prove that the alternated sum of the arithmetic Chern characters of these bundles is the sum of two terms, namely the secondary Bott Chern class of the sequence and its Chern character with support on the finite fibers.Next, we compute these classes in the situation encountered by the second author when proving a “Kodaira vanishing theorem” for arithmetic...

Given an elliptic curve E over a function field K = ℚ(T₁,...,Tₙ), we study the behavior of the canonical height $h{̂}_{E_{\omega }}$ of the specialized elliptic curve $E_{\omega }$ with respect to the height of ω ∈ ℚⁿ. We prove that there exists a uniform nonzero lower bound for the average of the quotient $\left(h{̂}_{E_{\omega }}\left(P_{\omega }\right)\right)/h\left(\omega \right)$ over all nontorsion P ∈ E(K).