2 changed title (finite-->finitely generated)

1

# Is a torsion free abelian group finite, if all of its localizations at primes p are finite over Zp?

Background: When proving that the group of -isogenies between two abelian varieties is finitely generated, one first shows that the Tate map is injective. Since each Tate module is free of finite rank over , it follows that the localization is -finite. One then uses a little trick to deduce the -finiteness of itself. (See Silverman I, for example.)

The above proof needs only a single prime . But disregarding issues of the characteristic of the field (which are apparently surmountable) we actually have an injective Tate map at every prime. Thus...

Question: Can the -finiteness of be deduced directly from the -finiteness of for all primes ?

One can consider this a question about general torsion-free abelian groups . A non-counterexample to keep in mind is , for which is -finite for all .

(A google search shows that there is actually quite a body of literature on torsion-free abelian groups, so perhaps the answer to this question is well-known, but I'm not sure where to look...)