Brauer-siegel for Arithmetic Tori and Lower Bounds for Galois Orbits of Special Points

Conjecture 1.1. Fix an integer g. Let B be a g-dimensional CM principally polarized abelian variety, and let x be the corresponding point in Ag,1(Q). Finally, let Z(End(B)) denote the center of the endomorphism ring of B. Then there exists a constant δg > 0 so that |Gal(Q/Q) · x| g Disc(Z(End(B)))g . Edixhoven established the desired lower bound for Hilbert modular surfaces in [3]. Recently, Pila [11] gave an unconditional proof of the André-Oort conjecture for an arbitrary product of modular curves. A positive answer to Conjecture 1.1 would be one of the main ingredients in generalizing Pila’s recent work to arbitrary Shimura varieties. Our main theorem is as follows: Theorem 1.1. For g ≤ 6, Conjecture 1.1 holds. If one assumes the Generalized Riemann Hypothesis for CM fields, then Conjecture 1.1 holds for all g ∈ N.