Abelian Varieties Having Purely Additive Reduction

Let E be an elliptic curve over a field K with a discrete valuation v with residue class field k. Suppose E has 'additive reduction' at o, i.e. the connected component A ° of the special fibre A0 of the N6ron minimal model is isomorphic to Q . Then the order of Ao(k)/A°(k) is at most 4 as can be seen by inspection of the usual tables, cf. [9, pp. 124-125] and [5, p. 46]. Thus it follows that if the order of the torsion subgroup Tors(E(K)) is at least 5 and prime to p = char(k), the reduction cannot be additive. This note arose from an attempt to see whether an explicit classification really is necessary to achieve this result. This attempt turned out to be successful: we prove a generalization for abelian varieties (cf. 1.15). The proof does nc use any specific classification, but it relies on monodromy arguments. It explains the special role of prime numbers l with l<_2g+ 1 in relation with abelian varieties of dimension g. Note that Serre and Tate already pointed out the importance of such primes, cf. [14, p. 498, Remark 2]. In their case, and in the situation considered in this paper the representation of the Galois group on T~A has dimension 2g, hence primes l with l _ 2g + 1 play a special role. We give the theorem and its proof in Section 1. Further we show that the bound in the theorem in sharp (Section 2), and we give examples in Section 3 which show that the restriction l :gchar(k) in the theorem is necessary. In Section 4 we indicate what can happen under the reduction map E(K)--'Eo(k) with points of order p in ca~ of additive reduction. Ribet made several valuable suggestions on an earlier draft of this paper. The elegant methods of proof in Section 2 were suggested by him. We thank him heartily for his interest in our work and for his stimulating remarks.