Research

Research Program

1. Number theory, especially elliptic curves. Arithmetic properties of elliptic curves over a number field or a function field, like the rank of the Mordell-Weil group, associated Galois representations, as well as applications to Diophantine equations, coding theory and arithmetic algebraic geometry.
2. Explicit methods in diophantine geometry, especially for rational points on higher genus curves and their Jacobians over number fields, with a focus on hyperelliptic and modular curves. Tools include classical and non-abelian Chabauty methods, heights and Arakelov theory, both classical and p-adic.
3. (Super)geometry and quantisation, with its focus on the (non)commutative geometry of Kontsevich's deformation- and Batalin-Vilkovisky's approaches to the quantisation of gauge field models.
4. Ordinary differential equations. This concerns geometric and algebraic, analytic aspects of linear differential and linear difference equations; differential Galois theory and its applications, in particular to symbolic (algorithmic) solvability of equations. We also study geometry of nonlinear differential equations, e.g., existence of algebraic solutions, and the Painlevé property.
5. Algebraic geometry related with curves, surfaces and threefolds: the maximal number of points on a curve of given genus over a given finite field; parametrizations of special rational surfaces, (history of) ruled surfaces, geometrical models, algebraic cycles; low genus curves with split Jacobians; reduction types of genus 3 curves.
6. Explicit methods in complex multiplication theory: classification of CM curves over number fields; construction of CM curves and abelian varieties; bounding the primes of bad reduction of genus-3 curves; construction of ray class fields over CM fields.

Part of our research is algorithmic, often leading to implementations in the computer algebra systems Sage or Magma.

Recent and current grants

• Rational Points: New Dimensions -- NWO-XL, joint with Martin Bright (Leiden), Valentijn Karemaker (Utrecht), Ronald van Luijk (Leiden), Marta Pieropan (Utrecht)
• Tropical methods in Mathematics and Computer Science -- Marie Skłodowska-Curie Individual Fellowship, joint with the Dynamical Systems, Geometry & Mathematical Physics group and the Fundamental Computing Science group.
• Global points via locally analytic functions -- NWO VIDI
• p-adic heights and p-adic BSD for modular abelian varieties with split purely toric reduction -- NWO Mathematics Cluster PhD position
• Quadratic Chabauty for integral points -- DFG Sachbeihilfe
• An explicit theory of heights for hyperelliptic Jacobians -- DFG Sachbeihilfe, joint with Michael Stoll (Bayreuth)