Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

Ribet has shown the following result: if $A$ is an abelian variety over a number field $E$, then the torsion subgroup of $A(E^{cyc})$ is finite. Here $E^{cyc}$ is the union $\bigcup_nE(\mu_n)$, $\mu_n$ being the set of n-th roots of unity.

Here we have used $E^{cyc}=E\mathbb{Q}^{ab}$. I would like to know if such finiteness result holds for other fields. For example, if $A$ is a CM abelian variety over a number field $E$, and let $F$ a totally real number field, with $F^{ab}$ the maximal abelian extension of $F$, can we expect the torsion subgroup of $A(EF^{ab})$ to be finite, too? Or should one expect the torsion subgroup of $A(EF^{ab})$ to be much smaller that the total torsion part of $A((EF)^{ab})$? The latter is infinite by a theorem of Zarhin, and I wonder if the torsion part can be somehow measured, and cut out some smaller subgroup.


share|improve this question
I like the non-standard question mark. –  darij grinberg Nov 27 '11 at 3:11

1 Answer 1

This is Corollary 2 in Show-Wu Zhang's Equidistribution of points of small heights on abelian varieties, Annals of Math. 147 (1998), no. 1, 159--165.

Observe that $\mathbf Q^{\mathrm{cyc}}=\mathbf Q^t(i)$, where $\mathbf Q^t$ is a totally real extension of $\mathbf Q$ (generated by all $\cos(\pi/n)$, $n>0$). Consequently, your assertion follows from the equidistribution theorem of Szpiro, Ullmo and Zhang, as applied to the Weil restriction of scalars $A'=R_{E(i)/\mathbf Q}A$. Indeed, if the set of torsion points of $A(E(i)\mathbf Q^t)$ were Zariski dense, one would be able to define a dense sequence of torsion points in $A'(\mathbf Q^t)$. The corresponding sequence of probability measures on $A'(\mathbf C)$ supported by their Galois orbits would be supported on $A'(\mathbf R)$ but converge (by the S-U-Z theorem) to the Haar measure on $A'(\mathbf C)$. The general case can be deduced from this one by passing to an appropriate abelian subvariety.

Note that Zhang attributes this result to Zarhin (Duke Math. J. 54 (1987)).

share|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.