Algebraization, transcendence, and D-group schemes

TL;DR

This paper proposes a conjecture linking algebraization, transcendence, and D-group schemes in Diophantine geometry, motivated by classical results and related to the Grothendieck Period Conjecture for cycles of codimension 1.

Contribution

It introduces a new conjecture connecting algebraization and transcendence via D-group schemes, and derives the Grothendieck Period Conjecture from classical transcendence theorems.

Findings

Formulation of a conjecture relating line bundles and D-group schemes.

Derivation of the Grothendieck Period Conjecture from transcendence theorems.

Connections established between algebraic geometry and transcendence theory.

Abstract

We present a conjecture in Diophantine geometry concerning the construction of line bundles over smooth projective varieties over $\bar{\mathbb Q}}$. 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 $\bar{\mathbb Q}}$. We also derive the Grothendieck Period Conjecture for cycles of codimension 1 in abelian varieties over $\bar{\mathbb Q}}$ from a classical transcendence theorem \`a la Schneider-Lang.