Orientable triangular embeddings of K18 − K3 and K13 − K3

K3 K3 K3 Surfaces with Involution and Analytic Torsion

In a series of works [Bo3-5], Borcherds developed a theory of modular forms over domains of type IV which admits an infinite product expansion. Such modular forms are said to be Borcherds's product in this paper. Among all Borcherds's products, Borcherds's Φ-function ([Bo4]) has an interesting geometric background; It is a modular form on the moduli space of Enriques surfaces characterizing the...

Metric packing for K3+K3

Exponentially many nonisomorphic orientable triangular embeddings of K12s+3

It was proved earlier that there are constants M, c > 0 such that for every n M (resp., every n M , n / ≡ 0, 3mod 12) there are at least c2n/6 nonisomorphic nonorientable (resp., orientable) genus embeddings ofKn. In the present paper we show that for s 6 there are at least 2s−6 nonisomorphic OT-embeddings of K12s . As a byproduct, we give a relatively simple method of constructing index four c...

Arithmetic of K3 Surfaces

Being surfaces of intermediate type, i.e., neither geometrically rational or ruled, nor of general type, K3 surfaces have a rich yet accessible arithmetic theory, which has started to come into focus over the last fifteen years or so. These notes, written to accompany a 4-hour lecture series at the 2015 Arizona Winter School, survey some of these developments, with an emphasis on explicit metho...

Arithmetic 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.