# Tate-Shafarevich groups of constant elliptic curves and isogeny volcanos

## TL;DR

This paper investigates the structure of Tate-Shafarevich groups for constant elliptic curves over function fields by analyzing the volcano structure of isogeny graphs over finite fields.

## Contribution

It introduces a novel approach connecting Tate-Shafarevich groups with isogeny volcano structures in elliptic curves over finite fields.

## Key findings

- Characterization of Tate-Shafarevich groups using volcano structures
- New insights into the relationship between isogeny graphs and arithmetic invariants
- Enhanced understanding of elliptic curves over function fields

## Abstract

We describe the structure of Tate-Shafarevich groups of a constant elliptic curves over function fields by exploiting the volcano structure of isogeny graphs of elliptic curves over finite fields.