# Mordell-Weil group of the universal abelian scheme

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 $(\ast)$.

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"Note that part of the difficulty of the question (∗) lies in the fact that I am asking for the structure of $\tilde{A}_{g,n}(K)$ 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
@Jason Starr: you are quite right. Thank you for pointing that out to me. This clearly simplifies the problem. –  Damian Rössler Aug 3 '12 at 12:27
Also you may be able to use Harer's work, which I believe proves the result when you restrict over the Torelli locus (orginally only in characteristic 0, but I think extended to char p by Schroeer). Of course we have injectivity of the restriction map from sections over all of the moduli space to sections over the formal completion of the Torelli locus. You may be able to use infinitesimal deformation theory to show that further restriction to the Torelli locus is also injective (ala SGA 2). –  Jason Starr Aug 3 '12 at 13:16
@Jason Starr. I am having difficulties finding the work by Harer you are referring to. Could you give me a more precise reference ? I like your idea to use deformation theory to restrict the problem to the Torelli locus. –  Damian Rössler Aug 3 '12 at 14:08
@Damian. I was referring to Harer's proof of the "Franchetta conjecture". This was extended by Schroeer to arbitrary characteristic. Here is the link to Schroeer's paper: http://reh.math.uni-duesseldorf.de/~schroeer/publications_pdf/franchetta.pdf` –  Jason Starr Aug 3 '12 at 15:27
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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.

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Dear Felipe, thank you very much for these references. Silverberg's article basically answers my question in char. $0$. If ever you come across a reference concerning the char. $p$ situation in higher dimensions, I would be grateful if you could send it to me. In fact, I initially asked myself what $\widehat{A}_{g,n}(K^{\rm perf})$ should be and then realized that the determination of $\widehat{A}_{g,n}(K)$ was already not easy. –  Damian Rössler Aug 3 '12 at 14:16
Note that the advertised result in this paper is the finiteness of the Mordell-Weil group. I just downloaded the paper, and she proves that it is torsion of the expected (i.e., minimal possible) order in one case: in Proposition 4.1 she shows it is (Z/nZ)^4. I would be interested in extensions of this work which completely determine the torsion subgroup, as in the papers of Shioda and Cox-Parry in the elliptic curve case. –  Pete L. Clark Sep 23 '13 at 5:32