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Algebraization, transcendence, and \(D\)-group schemes. (English) Zbl 1355.11074
Summary: We present a conjecture in Diophantine geometry concerning the construction of line bundles over smooth projective varieties over \(\overline{\mathbb{Q}}\). This conjecture, closely related to the Grothendieck period conjecture for cycles of codimension 1, is also motivated by classical algebraization results in analytic and formal geometry and in transcendence theory. Its formulation involves the consideration of \(D\)-group schemes attached to abelian schemes over algebraic curves over \(\overline {\mathbb{Q}}\). We also derive the Grothendieck period conjecture for cycles of codimension 1 in abelian varieties over \(\overline{\mathbb {Q}}\) from a classical transcendence theorem à la Schneider-Lang.

MSC:
11G25 Varieties over finite and local fields
11J81 Transcendence (general theory)
12H05 Differential algebra
14B20 Formal neighborhoods in algebraic geometry
14F40 de Rham cohomology and algebraic geometry
14K15 Arithmetic ground fields for abelian varieties
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