Complex multiplication and Brauer groups of K3 surfaces

Inspired by the classical theory of CM abelian varieties, in this paper we discuss the theory of complex multiplication for K3 surfaces. Let $X$ be a complex K3 surface with complex multiplication by the maximal order $\mathcal{O}_E$ of a CM field $E$. We compute the field of moduli of triples $(T(X), B, \iota)$, where $T(X)$ denotes the transcendental lattice of $X$, $B \subset \text{Br}(X)$ a finite, $\mathcal{O}_E$-invariant subgroup and $\iota \colon E \rightarrow \text{End}_{\text{Hdg}}(T(X)_{\mathbb{Q}})$ an isomorphism. If $X$ is defined over a number field $K$, we show how our results can be efficiently implemented to study the Galois-invariant part of the geometric Brauer group of $X$. As an application, we list all the possible groups that can appear as $\text{Br}(X)^{\Gamma_K}$ when $X$ has (geometric) maximal Picard rank, $K$ is the field of moduli of $(T(X)_{\mathbb{C}}, \iota)$ and $\Gamma_K$ denotes its absolute Galois group

Rational points in the Noether-Lefschetz locus of moduli spaces of K3 surfaces

In this paper, we study maps between moduli spaces of lattice-polarized K3 surfaces induced by sublattices of prime index. We show that these maps can be used to determine if a rational point of the moduli space belongs to the Noether-Lefschetz locus. As an application, we prove that the Bombieri-Lang conjecture implies non-density statements for the rational points in the Noether-Lefschetz locus, as predicted by a conjecture of Shafarevich.Comment: Improved exposition and result

THE BASAL COMPLEX STRATIGRAPHY OF THE HELMINTHOID MONTE CASSIO FLYSCH: A KEY TO THE EOALPINE TECTONICS OF THE NORTHERN APENNINES

Below the Monte Cassio helminthoid Flysch of the type locality, a well exposed basal complex outcrops in which Cenomanian turbidites of up to 200 m thickness are recognized. The bulk of these turbidites, named here Case Baruzzo Sandstone (CBS), consists of thick-bedded silty marlstones with a fine arenaceous base and represents the oldest siliciclastic input within the succession of the Cassio tectonic unit. The Case Baruzzo Sandstone lies unconformably on the Palombini shale of Hauterivian-Aptian age and on stratified packets of Jurassic-Cretaceous formations with Austroalpine affinity (Radiolarites, Aptici Shale and Maiolica). The Cenomanian CBS are unconformably overlain by Varicoloured Clay of Santonian - Campanian age and affected by soft-sediment deformations. The petrography of the CBS shows two petrofacies indicating (1) a direct provenance from their substrate and (2) an extrabasinal source similar to the terrigenous framework of the tectonically independent Coniacian-Santonian Ostia Sandstone outcropping southwest of the Cassio Unit. Because of its Cenomanian age the CBS must be considered as a siliciclastic wedge distinct from the younger Ostia Sandstone belonging to the Media Val Taro Unit and time correlative to the Varicoloured Clay of the Cassio Unit. The initiation of the turbidite sandstones terrigenous supply is Cenomanian into the Cassio Basal Complex (CBS), Coniacian into the Media Val Taro Unit (Ostia Sandstone) and Campanian into the more internal Gottero Unit (fine-grained turbidites interbedded within the Val Lavagna Formation). It is proposed that the relative positions of the highest tectonic units outcropping in the Emilian Apennines (i.e. Gottero, Media Val Taro and Cassio Units) during Late Cretaceous were not very different to the present setting, and that their tectono-stratigraphic evolution was related to Alpine-vergent accretionary wedges. The Alpine tectonic polarity should have controlled the westwards migration of the basinal depocentres and the evolution from the Adriatic-margin supply of the Cenomanian turbidites to the European-margin supply of the Maastrichtian turbidites. During the middle Eocene tectonic phase, before the counterclockwise apenninic rotation, the ophiolite-free Cretaceous Ligurian Units (i.e. Gottero, Media Val Taro, Cassio) overthrusted the Ligurian Units characterized by late Cretaceous ophiolitic detritus, known as Ottone and Caio Units, along a significant lithospheric discontinuity which acted as a transpressive fracture zone.&nbsp

The basal complex stratigraphy of the Helminthoid Monte Cassio Flysch: A key to the eoalpine tectonics of the Northern Apennines

Below the Monte Cassio helminthoid Flysch of the type locality, a well exposed basal complex outcrops in which Cenomanian turbidites of up to 200 m thickness are recognized. The bulk of these turbidites, named here Case Baruzzo Sandstone (CBS), consists of thick-bedded silty marlstones with a fine arenaceous base and represents the oldest siliciclastic input within the succession of the Cassio tectonic unit. The Case Baruzzo Sandstone lies unconformably on the Palombini shale of Hauterivian-Aptian age and on stratified packets of Jurassic-Cretaceous formations with Austroalpine affinity (Radiolarites, Aptici Shale and Maiolica). The Cenomanian CBS are unconformably overlain by Varicoloured Clay of Santonian - Campanian age and affected by soft-sediment deformations. The petrography of the CBS shows two petrofacies indicating (1) a direct provenance from their substrate and (2) an extrabasinal source similar to the terrigeneous framework of the tectonically independent Coniacian-Santonian Ostia Sandstone outcropping southwest of the Cassio Unit. Because of its Cenomanian age the CBS must be considered as a siliciclastic wedge distinct from the younger Ostia Sandstone belonging to the Media Val Taro Unit and time correlative to the Varicoloured Clay of the Cassio Unit. The initiation of the turbidite sandstones terrigenous supply is Cenomanian into the Cassio Basal Complex (CBS), Coniacian into the Media Val Taro Unit (Ostia Sandstone) and Campanian into the more internal Gottero Unit (fine-grained turbidites interbedded within thin the Val Lavagna Formation). It is proposed that the relative positions of the highest tectonic units outcropping in the Emilian Apennines (i.e. Gottero, Media Val Taro and Cassio Units) during Late Cretaceous were not very different to the present setting, and that their tectono-stratigraphic evolution was related to Alpine-vergent accretionary wedges. The Alpine tectonic polarity should have controlled the westwards migration of the basinal depocenters and the evolution from the Adriatic-margin supply of the Cenomanian turbidites to the European-margin supply of the Maastrichtian turbidites. During the middle Eocene tectonic phase, before the counter-clockwise apenninic rotation, the ophiolite-free Cretaceous Ligurian Units (i.e. Gottero, Media Val Taro, Cassio) overthrusted the Ligurian Units characterized by late Cretaceous ophiolitic detritus, known as Ottone and Caio Units, along a significant lithospheric discontinuity which acted as a transpressive fracture zone