Research

My research focuses on rational points, a topic at the crossroads of algebraic geometry and number theory. I study both geometric and analytic notions of abundance for rational points on varieties. I am particularly interested in the theory of semi-integral (Campana and Darmon) points, which forms a bridge between integral and rational points and provides a geometric perspective on solutions to equations with restricted prime factorisations. In my research, I employ tools from algebraic geometry, topology, analysis, Galois theory and number theory.

Hilbert property: I began my PhD by studying the Hilbert property, which is possessed by varieties whose sets of rational points are not thin. In a sense, this means that their rational points come from solving non-linear equations over the ground field, which is rare over arithmetically interesting fields. In [1], I showed that double conic bundles and certain del Pezzo surfaces (along with some higher-dimensional analogues) possess this property. This is connected to the inverse Galois problem, which asks if any finite group can be recognised as the Galois group of some field extension.

Weak approximation: Another geometric indicator of the abundance of rational points on a variety is weak approximation, a property possessed by varieties whose rational points over a number field are dense in the local points over all completions of the ground field. This property may be viewed as a geometric generalisation of the Chinese remainder theorem, and results in this area provide striking examples of the interplay between the "local" and "global" realms. In [3], Julian Demeio and I extended the methods used in [1] and combined them with a result of Denef relating arithmetic over local fields with the geometric property of splitness to prove that many del Pezzo surfaces of low degree satisfy a weak version of weak approximation, appropriately named weak weak approximation (WWA). This result offers support to a conjecture of Colliot-Thélène that all smooth unirational varieties satisfy WWA. In [6] (joint work with Julian Demeio and Rosa Winter), we extend geometric constructions used to prove unirationality of del Pezzo surfaces in order to prove WWA for all degree-two del Pezzo surfaces with a general rational point.

Semi-integral points: In [2], I established a Manin conjecture-type asymptotic for Campana points on orbifolds associated to norm forms for Galois extensions of number fields. This can be interpreted as an asymptotic for the number of powerful values of a norm form. The height zeta function approach I employed, modeled on the work of Batyrev—Tschinkel and Loughran, draws on both harmonic analysis and the theory of L-functions associated to Hecke characters. I am currently working with Alec Shute on vastly generalising this work.  In [4], Masahiro Nakahara and I explored the Hilbert property and weak approximation for Campana points, bringing together the seemingly disparate topics studied in my first two papers. Our work is the first to explore this topic in detail, despite its relevance to an important conjecture of Pieropan--Smeets--Tanimoto--Várilly-Alvarado, and we prove some foundational results. In [5] (joint work with Masahiro Nakahara and Vlad Mitankin), we studied both Campana points and a related notion which we have christened Darmon points (after Henri Darmon's work on M-curves), developing a semi-integral analogue of the Brauer-Manin obstruction and determining the status of local-global principles for orbifolds associated to quadric hypersurfaces.