COMPAUTGALREP · Computations of Automorphic Galois Representations

Computations of automorphic Galois representations

For over 350 years, Fermat's last theorem was the most exciting unresolved problem in Mathematics. The problem was finally resolved by Andrew Wiles in 1994. The Serre conjectures represent a vast extension of the ideas that went into the proof of Fermat's last theorem. They have played a central role in the development of number theory and were recently proved by Khare and Wintenberger. In essence, the Serre conjectures concern Galois representations of modular forms. A prerequisite to further theoretical progress in these directions is the development of tools for the explicit computation of Galois representations of modular forms. These tools will enable researchers to conduct experiments and formulate conjectures that will determine the direction of future research. This Marie Curie project has succeeded in developing some of these computational tools. This project resulted in two enhanced methods for the practical computation of Galois representations of modular forms: the first relies on floating-point approximations of torsion points of elliptic curves, and the second on p-adic approximations. These methods were developed by the researcher in collaboration with Peter Bruin (Zurich) and Maarten Derickx (Leiden). The project also resulted in a better understanding of the relation between the torsion subgroup of an elliptic curve and its Mordell-Weil rank through the discovery of the concept of 'false complex multiplication', in joint work with Peter Bruin (Zurich), Andrej Dujella (Zagreb) and Filip Najman (Zagreb). The project also resulted in the proof for the first time that several Groups of GL_2(F_q) type are in fact Galois groups, thus giving a positive answer to the famous Inverse Galois Problem for these groups. The fellow's website is: http://homepages.warwick.ac.uk/~maseap/Johan/index.html

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