Algebraization, Transcendence, and D-Group Schemes

Abstract

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.