Algebra and Number Theory

The research group carries out fundamental research on number theoretic aspects of Galois theory, elliptic curves, abelian varieties and modular forms, especially concerning Galois representations (modularity, congruences, local properties, images, compatible systems, independence), the inverse Galois problem, Kummer theory and local-to-global principles.
A particular feature is the development and the use of computer algebra tools for explicit study of examples and the compilation of databases. Experimental data often lead to unexpected observations and give the necessary insights to lead to a deeper theoretical understanding.

In 1637 Pierre de Fermat claimed that the equation aⁿ + bⁿ = cⁿ does not have any solution in positive integers a,b,c when n is at least 3. This statement, called Fermat's Last Theorem, remained unproved until 1994 and it is at the origin of the development of Algebraic Number Theory, i.e. the use of algebraic tools in number theory, as we know it nowadays. The proof that was finally found by Andrew Wiles, however, goes far beyond Algebraic Number Theory: it establishes deep links of Algebraic Number Theory with Geometry and Complex Analysis. More precisely, Wiles proved that, (certain) elliptic curves are parametrised by (certain) modular forms; since one knows how to associate an elliptic curve to a potential solution of the Fermat equation, one can take a look at the parametrising modular form and finds that it doesn't exist, whence the potential solution does not exist either.