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We present a conjecture in Diophantine geometry concerning the constructionof line bundles over smooth projective varieties over $\bar{\mathbb Q}}$. Thisconjecture, closely related to the Grothendieck Period Conjecture for cycles ofcodimension 1, is also motivated by classical algebraization results inanalytic and formal geometry and in transcendence theory. Its formulationinvolves the consideration of $D$-group schemes attached to abelian schemesover algebraic curves over $\bar{\mathbb Q}}$. We also derive the GrothendieckPeriod Conjecture for cycles of codimension 1 in abelian varieties over$\bar{\mathbb Q}}$ from a classical transcendence theorem \`a laSchneider-Lang.

Algebraization, transcendence, and D-group schemes