A finiteness conjecture on abelian varieties with constrained prime power torsion (Q1000635)

In the paper under review, the authors propose the following conjecture. Let \(K\) be a number field with fixed algebraic closure. For every prime \(\ell\), let \(K(\mu_\ell)\) be the field obtained from \(K\) by adjoining the \(\ell\)--th roots of unity, and let \(\tilde K_\ell\) be the maximal pro-\(\ell\) extension of \(K(\mu_\ell)\) which is unramified away from \(\ell\). Let \(g\geq 1\) be an integer and let \(A\) be an abelian variety of dimension \(g\) over \(K\). The field \(K(A[\ell^\infty])\), obtained from \(K\) by adjoining all \(\ell\)-th power roots of unity of \(A\), may or may not be contained in \(\tilde K_\ell\). If the inclusion \[ K(A[\ell^\infty]) \subseteq \tilde K_\ell \] indeed holds, then \(A\) has good reduction away from \(\ell\). It follows thus from \textit{G. Faltings}' results [Invent. Math. 73, 349--366 (1983; Zbl 0588.14026] that for fixed \(K\), \(g\) and \(\ell\), there are only finitely many isomorphism classes of abelian varieties for which the above inclusion holds. The authors conjecture that for fixed \(K\), \(g\), and for sufficiently big \(\ell\), there are in fact no abelian varieties of this kind. In other words: Conjecture: Let \(K\) be a number field and \(g\geq 1\) be an integer. There are only finitely many pairs \(([A], \ell)\), where \([A]\) is the isomorphism class of an abelian variety \(A\) of dimension \(g\) over \(K\) such that \(K(A[\ell^\infty]) \subseteq \tilde K_\ell\). For this conjecture, the following evidence is given. The authors prove (Theorem 2) that the statement of the conjecture holds in the case \(K=\mathbb Q\) (the field of rational numbers) and \(g=1\). The proof here relies on a result of Mazur characterizing noncuspidal rational points of modular curves. Also the case where \(K\) is a quadratic number field, not imaginary of class number one, and \(g=1\) is proven using now an extension of \textit{B. Mazur}'s theorem [Invent. Math. 44, 129--162 (1978; Zbl 0386.14009)] by \textit{F. Momose} [Compos. Math. 97, No. 3, 329--348 (1995; Zbl 1044.11582)]. Finally, the complete list of pairs \(([E], \ell)\), where \([E]\) is the isomorphism class of an elliptic curve \(E\) over \(K = \mathbb Q\) such that \(K(E[\ell^\infty]) \subseteq \tilde K_\ell\), is given.