Variations on the method of Chabauty and Coleman

One of the most important results in Diophantine geometry is the finiteness of the number of rational points on nice curves of genus at least two. However, there are no practical methods to compute the rational points on such curves in general, but the method of Chabauty and Coleman is one of the most successful approaches. We discuss variations on this method.

When applicable, the method of Chabauty and Coleman gives an upper bound on the number of rational points. We first study curves that attain this bound. Curves that attain the bound are rare and difficult to find; we construct several new examples.

When a curve satisfies a stronger rank condition, the method of Chabauty and Coleman can be extended to study points on symmetric powers of curves, which can tell us the information on points defined over small degree number fields. This method is called Symmetric Chabauty. We generalise the Symmetric Chabauty method so that one can use this extension to compute the cubic or quartic points on certain modular curves.

The limitation of the method of Chabauty and Coleman is the rank condition. There is a variation on the method that might be used in this case, called nonabelian Chabauty. When we use p-adic heights to construct a locally analytic function, a special explicit variant is called quadratic Chabauty. We give an algorithm to compute p-adic heights on even degree hyperelliptic curves and to apply quadratic Chabauty to compute integral points on certain even degree hyperelliptic curves.