**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 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 are moduli spaces for elliptic curves with additional data like level structure, e.g., isogenies of degreeexplicit and theoretical methods, for example using the Chabauty–Kim method
*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**.

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

Last modified: July 14, 2024

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