The Visibility Dimension

Let $A$ be an abelian variety over a field $K$ and fix $c\in H^1(K,A)$.

Definition 2.1   The visibility dimension of $c$ is the minimum of the dimensions of the abelian varieties $J$ such that $c$ is visible in $J$.

In Section 2.1 we prove an elementary lemma which, when combined with the proof of Proposition 1.3, gives an upper bound on the visibility dimension of $c$ in terms of the order of $c$ and the dimension of $A$. Then, in Section 2.2, we consider the visibility dimension in the case when $A=E$ is an elliptic curve. After summarizing the results of Mazur and Klenke on the visibility dimension, we apply a theorem of Cassels to deduce that the visibility dimension of $c\in {\mbox{{\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n}\selectfont Sh}}}(E)$ is at most the order of $c$.