# Elliptic surfaces to class groups and Selmer groups

## TL;DR

This paper explores the relationships between elliptic surfaces, class groups, and Selmer groups, revealing how torsion properties in these structures are interconnected in number theory.

## Contribution

It introduces new connections between the torsion of Picard groups, class groups, and Selmer groups in the context of elliptic surfaces.

## Key findings

- Link between n-torsion of Picard group and n-divisibility of class group
- Connection between Selmer group divisibility and class group divisibility
- Insights into the structure of torsion fields in elliptic surfaces

## Abstract

In this note, we connect the $n$-torsions of the Picard group of an elliptic surface to the $n$-divisibility of the class group of torsion fields for a given integer $n>1$. We also connect the $n$-divisibility of the Selmer group to that of the class group of torsion fields.