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 $(\ast)$. 



Mordell-Weil group of the universal abelian scheme
For $g=1$ there is a classical paper of Shioda (one of the two cited below) that proves that in char. zero, the group is what you expect but in char. p there are situations in which you get sections of infinite order.


On rational points of the generic elliptic curve with level N structure over the field of modular functions of level N. 
J. Math. Soc. Japan 25 (1973), 144–157


On elliptic modular surfaces. 
J. Math. Soc. Japan 24 (1972), 20–59. 


I seem to recall that in char zero the same is true for $g>1$ but I don't remember the reference, so I can't be sure. In char p, I don't know.


Added $g>1$ char zero:


Silverberg, Alice
Mordell-Weil groups of generic abelian varieties. 
Invent. Math. 81 (1985), 71–106. 


