Title: From Curves to Tropical Jacobians and Back,
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.