数学研究所

学术报告

数论研讨班

Speaker: Prof. Paolo Dolce（Westlake University）

Inviter: 黄治中 副研究员

Language: English

Title: The Arithmetic of Sections and Their Relative Monodromy in Abelian Schemes

Time&Venue: 2025年5月16日（星期五） 10:30-11:30 & 晨兴510

Abstract:  The Mordell-Weil group of abelian varieties over complex function fields can be studied using techniques from differential geometry and algebraic topology. Specifically, each rational point corresponds to an algebraic section of the associated abelian scheme. While such sections admit local logarithms, the obstruction to the existence of a global logarithm is encoded in a lattice known as the relative monodromy of the section. Intriguingly, this object appears to be deeply linked to the arithmetic properties of the section itself. For example, under certain hypotheses, the non-triviality of the relative monodromy is equivalent to the section being non-torsion. Furthermore, it is conjectured that the rank of the relative monodromy relates to the dimension of the minimal abelian subscheme containing the section’s image.In this talk, I will present partial results toward this conjecture, along with applications to transcendence problems. This is joint work with F. Tropeano (Università Roma Tre), extending earlier work of Corvaja and Zannier in the setting of elliptic surfaces.