Introduction

In this paper we give a survey of the problem of determining Mordell-Weil groups of universal abelian varieties.

We begin with the pioneering work of Shioda in the early 1970's on elliptic modular surfaces. In many ways, this is the most difficult and intriguing case. Shioda showed that the universal elliptic curve of level N in characteristic zero has only the N-torsion in its Mordell-Weil group over the field of elliptic modular functions of level N. The elliptic modular case is also the only case in which nontrivial examples are known of universal abelian varieties (in positive characteristic) with infinite Mordell-Weil group.

We then turn to abelian varieties (in characteristic zero) of higher dimension, paying special attention to two simple cases. The first, a direct generalization of the elliptic case, is that of the universal principally polarized abelian variety of dimension d > 1 and level N ≥ 3. Two proofs are sketched (Sections 2.2 and 2.7) of Shioda's conjecture that the Mordell-Weil group over the field of Siegel modular functions of level N is exactly the N-torsion (≅ = (Z/NZ)2d). These proofs are given as illustrative examples of some general techniques which were given in [12], [13], and [14]. The second case concerns abelian varieties whose endomorphism algebras contain an order in an indefinite division quaternion algebra over Q.