Research

How can one find the rational solutions to a polynomial equation? Some equations have no rational solutions at all, so a proof that none exist answers the question. In other cases, the question requires a way to describe the set of rational solutions.

For algebraic curves of genus at least two, there are only finitely many rational points, so one can list them together with a proof that the list is complete. In my research, I endeavour to make these questions accessible to explicit computation.

Below is a description of some of the main themes that have emerged in my research over the years.

Finite descent and Brauer–Manin obstructions

A curve over $\mathbb{Q}$ can have points over $\mathbb{R}$ and every $\mathbb{Q}_p$ and still have no rational points. Finite descent gives a way to detect some of these failures of the Hasse principle. Every rational point on the curve must lift to a rational point on one of finitely many twists of a chosen cover. If every twist on the list has a local obstruction, the original curve has no rational points. This is the idea behind explicit two-cover descent for hyperelliptic curves [Bruin–Stoll 2009] and for plane quartics [Bruin–Poonen–Stoll 2016]. For plane quartics whose 28 bitangents are all rational [Bruin–Lewis 2020], this method appears remarkably effective for proving that no rational points exist.

Descent obstructions have a close connection with Brauer–Manin obstructions. From that perspective, one can obtain a different formulation of finite descent obstructions that is much better suited to obtaining unconditional results in cases where the classical formulation usually requires the assumption of the generalized Riemann hypothesis to keep computations manageable [Bruin–Creutz 2026].

Chabauty methods and Mordell–Weil sieving

Once some rational points are known, how can one prove that the list is complete? When the rank of the Jacobian over $\mathbb{Q}$ is smaller than the genus, classical Chabauty uses $p$-adic analysis to prove that rational points cannot lie close together. Passing to a cover can make a Chabauty computation possible even when the rank condition fails on the original curve. In some cases a map to an elliptic curve over a number field makes the calculation practical, as in the determination of the primitive solutions to $x^2\pm y^4=\pm z^6$, $x^2 \pm y^4=z^5$, $x^2 \pm y^8 =z^3$, and $x^3+y^9=z^2$ [Bruin 1999b; Bruin 2003; Bruin 2005].

The Mordell–Weil sieve works in the other direction, constraining rational points to particular $p$-adic neighbourhoods by comparing the reductions of the curve and the rational points on its Jacobian at several primes.

With a combination of these methods, we have been able to decide the existence of rational points on a large collection of genus-two curves [Bruin–Stoll 2008; Bruin–Stoll 2010].

Mazur’s visibility of Tate–Shafarevich groups

Mazur suggested a particularly explicit approach to showing that certain varieties (homogeneous spaces of abelian varieties) have a local-to-global obstruction by realizing them as subvarieties of higher-dimensional abelian varieties. Such constructions require congruences on the $n$-torsion of abelian varieties. When one obtains these congruences, they can be used to make Tate–Shafarevich groups visible explicitly and obtain sharper bounds on the Mordell–Weil groups of abelian varieties. In various scenarios, my coauthors and I have been able to prove that such visibility arguments are guaranteed to work [Bruin 2004; Bruin–Flynn 2006; Bright–Bruin–Flynn–Logan 2007; Bruin–Dahmen 2010; Bruin–Fisher 2018].

Abelian varieties and moduli of curves

The world of hyperelliptic curves and their two-torsion is a rich source of examples of arithmetic phenomena, but their accessible structure hides the complexities that lie beyond. The three-torsion on abelian surfaces provides a stepping stone. The Burkhardt quartic threefold is birational to the moduli space of principally polarized abelian surfaces with full level-three structure. Over an open subset, one can write down a model of the universal genus-two curve. Twists of the threefold parametrize Kummer surfaces with prescribed level structure, and a Brauer class measures whether these Kummer surfaces come from abelian surfaces over the base field [Bruin–Nasserden 2018; Bruin–Filatov 2022]. We have also found applications of quotients of the Burkhardt quartic threefold that parametrize abelian surfaces with partial level structure [Bruin–Flynn–Testa 2014; Bruin–Flynn–Shnidman 2023].

The study of Jacobians of curves and their level structure rather naturally generalizes to the study of Prym varieties. These can still be understood through the geometry of curves, and they have interesting applications to the study of rational points on curves as well as to Mazur visibility [Bruin 2008; Bruin–Sertöz 2020; Bruin–Kulkarni 2025].

Rigorous numerical computation of period matrices

Abelian varieties over $\mathbb{C}$ allow for an entirely different description through complex analysis as complex tori. The translation between the algebraic and analytic realms goes through path integrals of algebraic functions. A careful study of such representations allows for the computation of the endomorphism ring of an abelian variety through numerical integration [Bruin–Sijsling–Zotine 2019; Bruin–Disney-Hogg–Gao 2024].

Hyperbolicity and symmetric differentials

Which algebraic surfaces contain only finitely many curves of geometric genus zero or one? This property is called algebraic quasi-hyperbolicity. Symmetric differentials on a resolution of a singular surface can constrain where such curves lie. On the surface parametrizing perfect cuboids, for example, every rational curve apart from certain known ones must pass through at least seven singular points [Bruin–Thomas–Várilly-Alvarado 2022].

For singularities of type $A_n$, one can compute their contribution to the Euler characteristic of the sheaves of symmetric differentials. These local computations give lower bounds on the number of global sections and help prove algebraic quasi-hyperbolicity for explicit surfaces of low degree [Bruin–Ilten–Xu 2025].

Generalised Fermat equations

In my PhD thesis [Bruin 1999a; Bruin 2002], I studied special cases of the Diophantine equation $x^r+y^s=z^t$. There is this remarkable theorem by Darmon and Granville.

The ABC-conjecture even suggests that all such solutions taken together (i.e., for varying $r,s,t$, counting $1^r+2^3=3^2$ as a single solution) should still form a finite set. There are some highly non-trivial solutions, though. The following list shows all known solutions. The small ones have been known for a long time, while the bigger ones were found by Beukers and Zagier (see Darmon and Granville, 1995, p. 515).

Since the ABC-conjecture suggests that the complete list is finite, one may wonder if the known solutions are actually all solutions. This would imply:

I established a partial result that handles all exponent triples that occur in the solution list above except $x^2+y^3=z^9$ (resolved in [Bruin 2005]) and $x^2+y^3=z^7$ (resolved by Poonen–Schaefer–Stoll in 2007):

The employed methods first show that such solutions correspond to rational points on algebraic curves of genus $g>1$. Then the rational points on those curves are obtained by Chabauty methods and covering techniques.