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Let $S$ be a Shimura variety and let $h$ be a Weil height function on $S$. Weconjecture that the heights of special points in $S$ are discriminantnegligible. Assuming this conjecture to be true, we prove that the sizes of theGalois orbits of special points grow as a fixed power of their discriminant (aninvariant we will define in the text). In the case of Shimura varieties ofabelian type, the height bound holds by the recently proved averaged Colmezformula, and our theorem gives a new proof of Tsimerman's Galois lower bound inthis case. The main novelty is that our approach avoids the use ofMasser-W\"ustholz isogeny estimates, replacing them by a point-countingargument, and establishes lower bounds for Galois orbits conditional on heightbounds for \emph{arbitrary} Shimura varieties. In particular, following thePila-Zannier strategy (and Gao's work in the mixed case) this implies that theAndre-Oort conjecture for an arbitrary (mixed) Shimura variety follows from thecorresponding conjecture on heights of special points.

Lower bounds for Galois orbits of special points on Shimura varieties: a point-counting approach