Differential characters of abelian varieties overp-adic fields

Geometry of Fermat adeles

“…We will first review some of the results in [5], [7] on differential characters, and results in [8], [1], [2] on (Siegel) differential modular forms. Next we will give two constructions, a crystalline one and a p−jet theoretical one of Siegel differential modular forms which we call f n cris,p and f n jet,p , respectively.…”

“…The following more precise statement follows from [5] Let U → B be a smooth morphism of smooth affine schemes over Z S . We say that U/B has special etale coordinates if …”

“…Let us quickly review the theory of p−jet spaces, as developed in [5], [8]. In what follows p is any prime integer.…”

“…Note also that the natural families of projections (J r+1 (Xˆp) → J r (Xˆp)) are morphisms in F Sch. Once we fix, for each X in Sch, a collection of isomorphisms (σ i ) as above one gets a functor Recall from [5] that for any r and any formal scheme X R , smooth over R ∈ Witt p , the projection J r (X R ) → X R is locally, in the Zariski topology, a trivial bundle with fiber (A nr )ˆp, the completion of the affine space of dimension nr, where n is the relative dimension of X R /R. (Here a formal scheme over R is called smooth if it is locally obtained by p−adically completing a smooth scheme over R. More generally a morphism of formal schemes will be called smooth if it is the p−adic completion of a smooth morphism of schemes of finite type over R.) Moreover these bundles for various r's are compatible with each other in the obvious sense.…”

“…For any p−adically complete ring M 0 let FSch M 0 be the full subcategory of the category of formal schemes over Spf M 0 whose objects are the formal schemes which are locally p−adic completions of schemes of finite type over M 0 . Given any M * ∈ Prol p , the theory in [5], [8] provides functors…”