THE abc THEOREM FOR COMMUTATIVE ALGEBRAIC GROUPS IN CHARACTERISTIC p

Buium proved what he called the abc theorem for abelian varieties over function fields in characteristic zero [3]. Using methods of algebraic model theory we prove an analog of his theorem for commutative algebraic groups in characteristic p. In what follows, k is an algebraically closed field of characteristic p, C is a smooth projective curve over k, and K = k(C) is the function field of C. Identify each point x ∈ C(k) with its corresponding valuation vx := ordx on K. The purpose of this paper is to demonstrate: Theorem 0.1. Let A be an abelian variety over K. Let f : A → P be a rational function. Let r be a positive integer, then there is a bound Br ∈ Z such that for any P ∈ A(K) either there are some a ∈ A(K) and Q ∈ A(K) such that f(Q) ∈ {0,∞} and P = Q+ [p]a or vx(f(P )) ≤ Br for any x ∈ C(k). Theorem 0.1 is the characteristic p analog of Buium’s abc theorem [3]. Our proof works for more general commutative algebraic groups and for distances computed to subvarieties of codimension greater than one. More general statements are in Section 1. Our proof follows the general form of Buium’s proof. We construct uniformly continuous homomorphisms from A(K) to some unipotent group with kernel [p]A(K) using differential algebra. The maps allow us to bound the distance from points in A(K)\[p]A(K) to 0. We then use a general lemma on approximations to reduce the theorem to the case of bounding the distance to a point. These estimates dovetail to give a proof of the theorem. Using Hrushovski’s Mordell-Lang theorem [8] we can give qualitative estimates on the growth of Br with r in many cases. The results of this paper formed a chapter of my Ph. D. thesis [14] written under the direction of E. Hrushovski whom I now thank for his advice. I thank B. Mazur for his advice and for insisting that a more geometric presentation of these arguments be given (though I admit that the argument is still not geometric). I thank D. Abramovich and J. F. Voloch for their comments on an earlier version. 1. A More General Formulation We give now a definition of the distance to a subvariety. Definition 1.1. Let Y ⊆ AK be a subvariety of affine space over K. Let P ∈ A(K). Let x ∈ C(k). Then the distance from P to Y is dvx(P, Y ) := min{vx(f(P )) : f ∈ IY ∩ OK,vx [X1, . . . , Xn]}