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 $g\geq 3$. Our proof uses Andr\'e's G-functions method, using formal and rigid uniformisation of semiabelian schemes to interpret the $p$-adic evaluations of the period G-functions.
Categories: math.NT math.AG