×

Points of finite order on abelian varieties. (English) Zbl 0787.14028

\(p\)-adic methods in number theory and algebraic geometry, Contemp. Math. 133, 175-193 (1992).
[For the entire collection see Zbl 0752.00052.]
Let \(A\) be an abelian variety of dimension \(d\) defined over a number field \(M\). The paper is mainly a survey article about explicit upper bounds of the order of \(A(M)\). Such upper bounds are given in the cases where:
(i) \(A\) has everywhere potentially good reduction.
(ii) \(A\) is of CM-type.
(iii) \(A\) has purely additive reduction at a discrete valuation \(v\) of \(M\).
The special case of elliptic curves is also considered.
In addition an explicit lower bound on ramification in extensions of number fields obtained by adjoining torsion on abelian varieties of CM- type is found.

MSC:

14K22 Complex multiplication and abelian varieties
14K15 Arithmetic ground fields for abelian varieties
11G10 Abelian varieties of dimension \(> 1\)
11G15 Complex multiplication and moduli of abelian varieties

Citations:

Zbl 0752.00052
PDFBibTeX XMLCite