Abelian varieties having purely additive reduction

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...

Abelian varieties having very bad reduction

Let E be an elliptic curve over a field K with a discrete valuation v with residue class field k. Suppose E has "very bad reduction" at v, i.e. the connected component An of the special fibre An of the Neron minimal model is isomorphic to ffi . Then U cl the order of An(k)/A„(k) is at most 4 äs can be seen by inspection of the usual tables, cf.[9], pp. 124/125, cf. [5], p.46. Thus it follows th...

Ordinary abelian varieties having small embedding degree

Miyaji, Nakabayashi and Takano (MNT) gave families of group orders of ordinary elliptic curves with embedding degree suitable for pairing applications. In this paper we generalise their results by giving families corresponding to non-prime group orders. We also consider the case of ordinary abelian varieties of dimension 2. We give families of group orders with embedding degrees 5, 10 and 12.

Semistable Reduction for Abelian Varieties

The semistable reduction theorem for curves was discussed in Christian’s notes. In these notes, we will use that result to prove an analogous theorem for abelian varieties. After some preliminaries on semi-abelian varieties (to convince us that the notion is a robust one), we will review the notion of a semi-abelian scheme (introduced in Christian’s lecture), recall the statement of the semista...

Good reduction of abelian varieties