Abstracts for ChaBONNty conference

In this talk, I will explain a somewhat new perspective on the relation between Coleman integration and Buium's theory of delta characters (which I will recall). I will then discuss applications to the Zilber--Pink conjecture, and possibly some generalisations. Parts of this are joint work with Arnab Saha (IIT Gandhinagar) and Sudip Pandit (King's College London).

Mazur's celebrated theorem, together with follow-up work by Kenku and Momose, gives a complete classification of the rational points on the modular curves $X_0(N)$. This result is special to $\mathbb{Q}$, and no direct analogue holds over general number fields. In this talk, we restrict to the case of quadratic number fields $K$ and survey the current state of the art for understanding $X_0(N)(K)$. We describe the role of Atkin-Lehner involutions and Chabauty-type methods that together allow one to determine quadratic points on $X_0(N)$ in some cases.



 

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.