Research topics

Arithmetic geometry, especially arithmetic and computational aspects of curves and abelian varieties over arithmetic fields, their rational points, L-functions, and cohomology

• abelian varieties and the Birch–Swinnerton-Dyer conjecture:
  The Birch–Swinnerton-Dyer (BSD) conjecture states a surprising and deep relation between algebraic and analytic, local and global invariants of elliptic curves over Q, and more generally of abelian varieties over global fields. There are many important consequences, for example the existence of an algorithm to compute the Mordell–Weil group.
  • over number fields: explicit methods and Iwasawa theory
  • in positive characteristic: over higher-dimensional bases
• rational points on (modular) curves:
  Modular curves are moduli spaces for elliptic curves with additional data like level structure, e.g., isogenies of degree N. Knowing their rational points is important for many other questions in arithmetic geometry. For example, they allow one to classify the possible torsion subgroups of elliptic curves over Q.
  explicit and theoretical methods, for example using the Chabauty–Kim method

You can find the code for my articles on GitHub when they are published.