# From Curves to Tropical Jacobians and Back

## TL;DR

This paper explores the relationship between algebraic curves and their tropical Jacobians, providing methods for tropicalizing curves, computing their Jacobians, and reconstructing curves from tropical data, with a focus on hyperelliptic cases.

## Contribution

It introduces a new approach for hyperelliptic curves to find their tropicalizations and Jacobians, and discusses algorithms for reconstructing curves from tropical period matrices.

## Key findings

- Developed a method for tropicalizing hyperelliptic curves using admissible covers.
- Described how to compute the tropical Jacobian and theta divisor from a weighted metric graph.
- Addressed the problem of reconstructing algebraic curves from tropical period matrices.

## Abstract

Given a curve defined over an algebraically closed field which is complete with respect to a nontrivial valuation, we study its tropical Jacobian. This is done by first tropicalizing the curve, and then computing the Jacobian of the resulting weighted metric graph. In general, it is not known how to find the abstract tropicalization of a curve defined by polynomial equations, since an embedded tropicalization may not be faithful, and there is no known algorithm for carrying out semistable reduction in practice. We solve this problem in the case of hyperelliptic curves by studying admissible covers. We also describe how to take a weighted metric graph and compute its period matrix, which gives its tropical Jacobian and tropical theta divisor. Lastly, we describe the present status of reversing this process, namely how to compute a curve which has a given matrix as its period matrix.