Let $n>2$ and let $k$ be either $\bf Q$ or a finite field whose characteristic is prime to $n$. Let $A_{g,n}$ be the moduli scheme, which represents the functor, which with every $k$-scheme $S$ associates the set of principally polarized abelian schemes $\cal A$ over $S$, together with a symplectic isomorphism $({\bf Z}/n{\bf Z})^{2g}_S\simeq A[n]$.` This scheme is geometrically irreducible by Chai-Faltings. Let ${\widetilde A}_{g,n}\to A_{g,n}$ be the universal family and let $K$ be the function field of $A_{g,n}$.

My ${\bf question}$ is: is anything known about ${\widetilde A}_{g,n}(K)$ ? $(\ast)$

A guess would be that ${\widetilde A}_{g,n}(K)\simeq {\widetilde A}_{g,n}[n](K)$.`

Note that part of the difficulty of the question $(\ast)$ lies in the fact that I am asking for the structure of ${\widetilde A}_{g,n}(K)$ and not for the structure of its subset ${\widetilde A}_{g,n}({A}_{g,n})$.`

In the case $g=1$ (elliptic curves), these two sets coincide and the question should be easier to answer.

A final remark is that question $(\ast)$ is maybe not "the right one". It might make more sense to ask for the structure of the group of rational sections of the universal abelian scheme over the moduli stack of all abelian varieties (forgetting level structures and even polarizations) - but this group is not the Mordell-Weil group of a concrete abelian variety so I prefer to focus on the more down-to-earth question $(\ast)$.

`$\tilde{A}_{g,n}(K)$`

and not for the structure of its subset`$\tilde{A}_{g,n}(A_{g,n})$`

." -- By the Weil extension property these are the same (at least when $n>2$ so that there is a fine moduli space). – Jason Starr Aug 3 '12 at 12:14`http://reh.math.uni-duesseldorf.de/~schroeer/publications_pdf/franchetta.pdf`

– Jason Starr Aug 3 '12 at 15:27