Joint IAS/PU Arithmetic Geometry

Given a family $g: X \rightarrow S$ of smooth projective algebraic varieties over a number field $K$, one often wants to constrain the points $s$ in $S$ where the fibre $X_s$ acquires "extra" algebraic structure. A basic sort of constraint which is important in unlikely intersection theory is that of a Galois-orbit lower bound: an inequality $h(s) \le poly([K(s) : K])$, where $h$ is some logarithmic Weil height and $K(s)$ is the field of definition of $s$. Recent work has focused on how to use $G$-functions constructed from degenerations of $g$ to produce such inequalities.

We describe some new results in the case where $g$ is a one-parameter degeneration of surfaces, and the central role played by rigid and "adelic" geometry.