Local heights away from $p$ for quadratic Chabauty

In this talk, we present an algorithm for computing Nekovář local heights at primes $\ell \neq p$ on hyperelliptic curves, a key ingredient in quadratic Chabauty. The method builds on work of Betts and Dogra, which relates local heights to the geometry of a semistable model of the curve and the action of correspondences. Using cluster pictures and the Coleman--Iovita isomorphism, we obtain an effective procedure for carrying out these computations in practice. We discuss applications to quadratic Chabauty, including an example with non-trivial local height contributions at two primes, and report on ongoing work extending these ideas beyond the hyperelliptic setting. This is joint work with Alexander Betts, Sachi Hashimoto, and Pim Spelier.