Points of small order on three-dimensional abelian varieties; with an appendix by Yuri Zarhin
If \(A\) is an abelian variety over a number field \(K\), and \(A_\wp\) has an \(\ell\)-torsion point (\(\ell\) prime) for almost all primes \(\wp\) of \(K\), does some \(A'\), isogenous over \(K\) to \(A\), have an \(\ell\)-torsion point? In general the answer is no (though it is yes for surfaces) but for \(3\)-folds the exceptions are few enough to be worth classifying, and that has been done for \(\ell\geq 7\) by the author. For \(\ell=2\) the answer is yes for \(3\)-folds, so what remains is \(\ell=3\) and \(\ell=5\). This is what is dealt with here: more precisely, the object of study is the image of the mod \(\ell\) representation \[ \overline\rho_\ell: {\mathop{\mathrm{Gal}}}(\overline K/K)\to {\mathop{\mathrm{Aut}}}(T_\ell(A)\otimes{\mathbb F}_\ell)\cong {\mathop{\mathrm{GL}}\nolimits}_6({\mathbb F}_\ell). \] The main result is that if \(A'\) does exist then the Levi component of the image is \({\mathbb Z}/2\times {\mathbb Z}/2\), \(D_4\), \(S_4\) or (for \(\ell=5\)) \(D_5\). The problem was reformulated in those terms by N.~Katz, and the Weil pairing allows us to consider subgroups of symplectic groups only. So most of the proof consists of an analysis of the maximal subgroups of \({\mathop{\mathrm{Sp}}\nolimits}_6({\mathbb F}_3)\) and \({\mathop{\mathrm{Sp}}\nolimits}_6({\mathbb F}_5)\), which are known, and showing that most of them do not contain an appropriate subgroup. Zarhin's appendix in concerned with \(\ell=2\) and \(2\leq \dim A\leq 4\). In those cases, if \(A_2\) is an absolutely simple simple module for the image of the Galois group in \({\mathop{\mathrm{Aut}}} A_2\) then \({\mathop{\mathrm{End}}}A={\mathbb Z}\) or \(K\) is of positive characteristic and \(A\) is supersingular.
scientific article; zbMATH DE number 5768898
scientific article