arXiv subjects

Francesco Tropeano

Publications and source records attributed to Francesco Tropeano.

4 recordsLinked to original sources

arXiv2025

Relative monodromy of ramified sections on abelian schemes

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.

math.NT↗

arXiv2025

Finite translation orbits on double families of abelian varieties (with an appendix by E. Amerik)

We study two families of $g$-dimensional abelian varieties, induced by distinct rational maps defined on a common variety $\overline{\mathcal A}$ and mapping to two bases $\overline{S}_1$ and $\overline{S}_2$. Two non-torsion sections induce birational fiberwise translations on $\overline{\mathcal A}$. We consider the action of a specific subset of the group generated by these translations. Under the assumption that $\operatorname{dim} \overline{S}_1 (= \operatorname{dim} \overline{S}_2) \leq g$, we prove that the points with finite orbit are contained in a proper Zariski closed subset. This subset is explicitly described to a certain extent. Our results generalize a theorem of Corvaja, Tsimermann, and Zannier to higher dimensions.

math.NT↗

arXiv2024

Monodromy of elliptic logarithms: some topological methods and effective results

We present some effective approaches in studying the relative monodromy group of elliptic logarithms with respect to periods of elliptic schemes. We provide explicit ways of constructing explicit loops which leave periods unchanged but along which logarithms have non-trivial variations. We also get some topological methods and effective results which allow to manage the ramification locus of sections. The paper was inspired by a theorem of Corvaja and Zannier which abstractly determine the relative monodromy group of non-torsion sections.

math.NT↗

arXiv2023

Monodromy of double elliptic logarithms

We determine the relative monodromy group of abelian logarithms with respect to periods in the cases of fibered products of elliptic schemes. This gives rise to a result stronger than a theorem due to Y. André and implies in particular the algebraic independence of the logarithm of any non-torsion section and the periods. We then conjecture an analogous result for the general case of an abelian scheme of arbitrary relative dimension. This generalizes a theorem of Corvaja and Zannier which determines the said group in the case of a single elliptic scheme.

math.NT↗