Quadratic Chabauty on $X_0(N)^*$, Part 1: Semistable models

In order to carry out the quadratic Chabauty method for determining rational points on curves, one needs to be able to compute height contributions at the primes of bad reduction. These height contributions can be calculated in terms of a semi-stable model of the curve.

In this talk, we explicitly describe the semi-stable models of $X_0(N)^*$ with $N$ square-free. In the quadratic Chabauty method, there is some freedom of choice of a correspondence. As an application of our description, we will give an explicit description of how to choose this correspondence so that all height contributions at the primes of bad reduction are 0, removing the need to compute these contributions! Choosing this correspondence to have trivial height contributions is not always possible; the question of when it exists will be part of the second talk.