Relative monodromy of ramified sections on abelian schemes
Paolo Dolce, Francesco Tropeano
TL;DR
We address the obstruction to a globally defined abelian logarithm for ramified sections of a complex abelian scheme without fixed part and with maximal variation. The central object is the relative monodromy $M^{\mathrm{rel}}_\sigma$, which the paper shows is nontrivial and, under irreducible period monodromy, equals $\mathbb Z^{2g}$, with extensions to certain non-torsion cases. The authors develop a Lefschetz-hyperplane strategy and trace arguments to derive these results and apply them to reprove Manin's kernel theorem and to obtain algebraic-independence results for abelian logarithms in the Pila–Zannier setting. These findings tie the geometry of monodromy to the Betti map and functional transcendence, offering new tools for unlikely-intersections problems and special-point questions in abelian-family contexts.
Abstract
Let's fix a complex abelian scheme $\mathcal A\to S$ of relative dimension $g$, without fixed part, and having maximal variation in moduli. We show that the relative monodromy group $M^{\textrm{rel}}_σ$ of a ramified section $σ\colon S\to\mathcal A$ is nontrivial. Moreover, under some hypotheses on the action of the monodromy group $\textrm{Mon}(\mathcal A)$ we show that $M^{\textrm{rel}}_σ\cong \mathbb Z^{2g}$. We discuss several examples and applications. For instance we provide a new proof of Manin's kernel theorem and of the algebraic independence of the coordinates of abelian logarithms with respect to the coordinates of periods.