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Given a point $\xi$ on a complex abelian variety $A$, its abelian logarithmcan be expressed as a linear combination of the periods of $A$ with realcoefficients, the Betti coordinates of $\xi$. When $(A, \xi)$ varies in analgebraic family, these coordinates define a system of multivaluedreal-analytic functions. Computing its rank (in the sense of differentialgeometry) becomes important when one is interested about how often $\xi$ takesa torsion value (for instance, Manin's theorem of the kernel implies that thiscoordinate system is constant in a family without fixed part only when $\xi$ isa torsion section).  We compute this rank in terms of the rank of a certain contracted form of theKodaira-Spencer map associated to $(A, \xi)$ (assuming $A$ without fixed part,and $\mathbb{Z} \xi$ Zariski-dense in $A$), and deduce some explicit lowerbounds in special situations. For instance, we determine this rank in relativedimension $\leq 3$, and study in detail the case of jacobians of families ofhyperelliptic curves.  Our main application, obtained in collaboration with Z. Gao, states that if$A\to S$ is a principally polarized abelian scheme of relative dimension $g$which has no non-trivial endomorphism (on any finite covering), and if theimage of $S$ in the moduli space $\mathcal{A}_g$ has dimension at least $g$,then the Betti map of any non-torsion section $\xi$ is generically asubmersion, so that $\xi^{-1}A_{tors}$ is dense in $S(\mathbb{C})$.

The Betti map associated to a section of an abelian scheme (with an appendix by Z. Gao)