# Abelian variety over Q with many roots of unity

Given an abelian variety $$A$$ over the rational integers $$\mathbb{Q}$$, and a prime $$p$$, we know that $$\mathbb{Q}(\zeta_p)$$ is contained in $$\mathbb{Q}(A[p])$$, the $$p$$-division field of $$A$$, and where $$\zeta_p$$ is the primitive $$p$$-th root of unity.

More generally, for a finite set $$S\subset A(\bar{\mathbb{Q}})$$, let denote by $$\mathbb{Q}(S)$$ the field obtained by adjoining to $$\mathbb{Q}$$ the coordinates of the points in $$S$$.

We know also that $$A[p](\bar{\mathbb{Q}})\cong (\mathbb{Z}/p\mathbb{Z})^{2g}$$, where $$g$$ is the dimension of $$A$$.

Can we have a simple abelian variety $$A$$ over $$\mathbb{Q}$$ such that for each odd prime $$p$$, we have at most one subgroup $$G\subset A[p](\bar{\mathbb{Q}})$$ of order $$p$$ such that $$\zeta_p\notin \mathbb{Q}(G)$$?