On relative integral monodromy of abelian logarithms and normal functions

TL;DR

This paper investigates the relative integral monodromy of abelian logarithms and normal functions, showing it forms a lattice under certain conditions and providing a Hodge-theoretic interpretation.

Contribution

It establishes conditions under which the relative integral monodromy is a lattice in the algebraic monodromy, extending previous results to a broader context.

Findings

Relative integral monodromy is a lattice when the abelian scheme's integral monodromy is a lattice.

The work provides a Hodge-theoretic interpretation of sections of abelian schemes.

Generalizes the concept of relative monodromy to normal functions.

Abstract

The relative algebraic monodromy of abelian logarithms (defined as the kernel of a map between algebraic monodromy groups attached to an abelian scheme with and without a section) was computed in \cite{A1}: under natural assumptions, this vector group turns out to be maximal. The relative integral monodromy of abelian logarithms is defined similarly as a kernel of integral monodromy groups, without taking Zariski closures. We show that if the integral monodromy of the abelian scheme is a lattice in the algebraic monodromy (which is not always the case), then the relative integral monodromy of the abelian logarithm is also a lattice in the relative algebraic monodromy. The proof uses a Hodge-theoretic interpretation of sections of abelian schemes. We also consider relative integral monodromy groups in the more general context of normal functions.