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TORSION POINTS OF ABELIAN VARIETIES OVER FUNCTION FIELDS

Let C be a smooth projective geometrically connected curve defined over a finite field k = Fq of characteristic p, K = k(C) its function field of genus g and A/K a non-constant abelian variety of dimension d. Let φ : A → C be a Néron model of A over C. The ultimate goal is to show that if the Kodaira-Spencer map of φ is non-zero and p > 2d + 1, then there exists a bound for the order of the torsion subgroup of A(K) depending only on d and g (which alas is not explicit). The proof requires several steps.