Relative monodromy of ramified sections on abelian schemes

TL;DR

This paper investigates the monodromy properties of ramified sections on complex abelian schemes, establishing nontriviality and specific isomorphisms of the relative monodromy group, with applications to classical theorems.

Contribution

It proves the nontriviality of the relative monodromy group for ramified sections and characterizes it as isomorphic to g under certain conditions, providing new insights and proofs.

Findings

The relative monodromy group of a ramified section is nontrivial.

Under certain hypotheses, the monodromy group is isomorphic to g.

Provides new proofs of Manin's kernel theorem and algebraic independence results.

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 $M^{\textrm{rel}}_\sigma$ of a ramified section $\sigma\colon S\to\mathcal A$ is nontrivial. Moreover, under some hypotheses on the action of the monodromy group $\textrm{Mon}(\mathcal A)$ we show that $M^{\textrm{rel}}_\sigma\cong \mathbb Z^{2g}$. 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.