Any complex abelian variety \(A\) of dimension \(g\) can be represented as a complex torus \({\mathbb{C}}^g/{\mathcal L}\simeq ({\mathcal L}\otimes_{\mathbb{Z}}{\mathbb{R}})/{\mathcal L}\) and every point \(\xi\) of \(A\) may be identified by its \(2g\) real coordinates in a mesh of the lattice \({\mathcal L}\). When \((A,\xi)\) moves in an algebraic family, one gets the Betti map \(\beta\) from the parameter space \(S\) to \({\mathbb{R}}^{2g}\), which is a multivalued analytic map. This is a convenient tool for the study of the distribution of torsion values. The authors list a sample of classical or recent occurrences of torsion value problems and/or Betti maps, and undertake a systematic study of the Betti map. They relate the derivative of the Betti map to the Kodaira-Spencer map. Under natural hypotheses, they show that the Betti map of every section not contained in an algebraic subgroup is a submersion, and deduce the density of torsion values for every section.
Let \(A\to S\) be an abelian scheme of relative dimension \(g\) over a smooth complex algebraic variety, \(\xi:S\to A\) a section, \(\widetilde{S}\) the universal covering of \(S({\mathbb{C}})\), \(\widetilde{\beta}:\widetilde{S}\to {\mathbb{R}}^{2g}\) the associated Betti map. The authors give formulae for the generic rank \({\mathrm{rk}}\beta\) of \(\beta\), namely the maximal value of the rank of the derivative \(d\beta(\tilde{s})\) when \(\tilde{s}\) runs through \(\tilde{S}\). The authors determine this rank in relative dimension \(\le 3\) and investigate in detail the case of jacobians of families of hyperelliptic curves, both in the real and complex case. The main application is obtained in collaboration with Z. Gao. Let \(A \to S\) be a principally polarized abelian scheme of relative dimension \(g\) which has no non-trivial endomorphism (on any finite covering). Assume that the image of \(S\) in the moduli space \(A_g\) has dimension at least \(g\). Then the Betti map of any non-torsion section \(\xi\) is generically a submersion, so that \(\xi^{-1} A_{\mathrm {tors}}\) is dense in \(S({\mathbb C})\). The proof involves an application of the pure Ax-Schanuel theorem written by Z. Gao in an Appendix to the paper under review (see [N. Mok et al., Ann. Math. (2) 189, No. 3, 945–978 (2019; Zbl 1481.14048)]).