Algebra and Number Theory

The research group carries out fundamental research on number theoretic aspects of Galois theory, modular forms, elliptic curves and abelian varieties, especially concerning Galois representations (modularity, congruences, local properties, images, compatible systems, independence), modular forms and the inverse Galois problem.
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. The research group is involved in the Priority Program on Computer Algebra through an FNR/DFG-Inter grant.

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.

 

The parametrisation of elliptic curves by modular forms found by Wiles has nowadays been widely generalised into a parametrisation of (certain) Galois representations by modular forms. The absolute Galois group can be considered as the symmetry group of all solutions to equations with rational coefficients. It encodes the essential information on all rational equations and is thus one of the main objects in number theory and geometry. However, it is huge (profinite, having uncountably many elements) and its entire understanding is currently totally out of reach. Like any group in Physics, Chemistry and Mathematics, it is natural to study it through its representations, these are the so-called Galois representations. Such Galois representations are provided, for instance, by modular forms, elliptic curves and abelian varieties. These objects are very important beyond number theory, namely they are essential tools in modern cryptography.

The Algebra and Number Theory group works in this very active area and contributes to the understanding of certain new aspects in these wide theories. For more details, please refer to the description of the research projects.