- 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, the conductor, associated Galois representations, as well as applications to Diophantine equations, coding theory and arithmetic algebraic geometry, are the objects of study.
- Ordinary differential equations. This concerns algebraic, analytic (e.g., mul- tisummability) and algorithmic aspects of linear differential and linear differ- ence equations; differential Galois theory and its applications, in particular to symbolic (algorithmic) solvability of equations; differential equations having the Painlevé property.
- 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.
- Drinfeld modules. This concerns a theory in positive characteristic which has similarities with the theory of elliptic curves and Abelian varieties.
Overview of scientific results
See the list of publications.