Cyclic components of abelian varieties (\(\operatorname{mod} \wp\)) (Q746951)

Let \(A\) be an abelian variety of dimension \(r\) defined over a number field \(F\) and let \(\mathcal P\) be a prime of good reduction. The group of \({\mathbb F}_{\mathcal P}\)-points of the reduced variety is isomorphic to the product of cyclic groups \({\bar A}({\mathbb F}_{\mathcal P})\cong {\mathbb Z}/m_{1}{\mathbb Z} \times \cdots \times {\mathbb Z}/m_{s}{\mathbb Z} \) where \(s\leq 2r, \,\, m_{i}\geq 2 \) and \(m_{i} | m_{i+1},\,\, 1\leq i \leq s-1.\) If \(s<2r\) then one says that \({\bar A}({\mathbb F}_{\mathcal P})\) has at most \((2r-1)\) cyclic components. Let \[ f_{A,F}(x)=|\{{\mathcal P}\in {\mathcal P}_{A} : N_{F/{\mathbb Q}} {\mathcal P} \leq x, \,\, {\bar A}({\mathbb F}_{\mathcal P}) \,\, {\text{has at most }} \, (2r-1) \, {\text{cyclic components}} \}|. \] Here, \({\mathcal P}_{A}\) denotes the set of primes of good reduction for \(A\) and \(N_{F/{\mathbb Q}} \) is the norm map. Let \(F(A[m])\) be the field obtained by adjoining to \(F\) the \(m\)-division points \(A[m]\) of \(A.\) The main result of the paper, under the assumption of the generalized Riemann hypotheses for the Dedekind zeta functions of division fields for \(A,\) is the following estimate for the function \(f_{A,F}:\) \[ f_{A,F}(x)=c_{A,F}\, {\text{li}}(x) + O(x^{5/6}({\text{log}} x)^{2/3}), \] where \({\text{li}}(x)=\int_{2}^{x}\frac{1}{{\text{log}}\, t},\) \(c_{A,F}=\sum_{m=1}^{\infty}\frac{{\mu}(m)}{[F(A[m]):F]}\) and \({\mu}\) is the Möbius function.