The large Galois orbits conjecture under multiplicative degeneration

We establish the PEL type large Galois orbits conjecture for Hodge generic curves in ${\mathcal{A}}_g$ possessing multiplicative degeneration. Combined with our earlier works, this concludes the proof of the Zilber-Pink conjecture in ${\mathcal{A}}_2$ for such curves. We also deduce several new cases of Zilber-Pink in ${\mathcal{A}}_g$ for $g\ge 3$. Our proof uses André’s G-functions method, using formal and rigid uniformisation of semiabelian schemes to interpret the $p$-adic evaluations of the period G-functions.