Elements of Order <i>P</i> in the Tate-šafarevič Group

Abstract

By a global field, we will always mean a number field or a function field in one variable over a finite field.Lang and Tate have shown [4] that the Tate-Safarevic" group of an abelian variety over such a field has only finitely many elements of order dividing a fixed integer m, provided that m is not divisible by the characteristic p of the field.The purpose of this note is to remove the restriction on m.Our proof is complicated by the fact that, unlike the prime-to-p case, there may be an infinite number of principal homogeneous spaces of order p over the global field which split locally at all primes outside a given non-empty finite set, and hence we must work with all the primes of K. THEOREM. For any abelian variety A over a global field K, and any integer m, the Tate-Safarevic group \ \ \ (A/K) of A over K has only finitely many elements of order dividing m.After [4; Theorem 5], we may assume that K has non-zero characteristic p, and that m = p.As in the proof of the above-cited theorem, we may replace K by a finite separable extension and hence assume that the kernel of the isogeny p : A -* A (as a finite group scheme) has a composition series whose quotients are one of the three group schemes Z/pl, cn p , \i p [6].Correspondingly, p : A -> A will be a composite of isogenies, p = $2d°02d-i°-••°0i> $ i : ^f -i -* A h with all the <j) { of degree p.Thus, it suffices to prove the statement: let 4>: A -> B be an isogeny over K with kernel equal to one of I/pZ, a p , or ji p .Then the kernel of the map induced by $ is finite.We will write H'(S, -) for a cohomology group with respect to the flat (/.p. q.f.) topology on S (or spec S if S is a ring), X for the complete, smooth algebraic curve canonically associated to K/k, and X o for the set of closed points of X (i.e.primes of K).If v is in X o , then K v is the completion of K at v and R v the ring of integers in K u .All of our group schemes will be commutative.If G is a group scheme of finite type over K v , then there is a canonical topology on H l (K v , G) [6].Let Jf be a finite flat group scheme over R v and let N = Jf ® Rv K v .Then the canonical map H*(R V , Jf) -* H i (K v ,N) is injective, because a principal homogeneous space for Jf over R v which has a point in K v clearly already has a point in R v .Thus, any element of H 1 (R V , J/") is split by the integral closure R v ' of

Locations

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