Periodic Reporting for period 1 - GalRepsDiophantine (Galois Representations and Diophantine Problems) Reporting period: 2018-03-01 to 2020-02-29 Summary of the context and overall objectives of the project For over 350 years Fermat's Last Theorem was the most famous open problem in mathematics, and was finally resolved by Andrew Wiles in 1994. Whilst Wiles' proof had dramatically succeeded in resolving the Fermat equation over the rationals, for many other Diophantine problems (including the Fermat equation over number fields), the proof strategy is insufficient. Indeed, the approach in Wiles' proof, building on ideas of Frey, Serre and Ribet, associates a putative solution of certain Diophantine equations to a Frey elliptic curve, and then predicts that the residual Galois representation of that elliptic curve comes from a finite computable set of modular Galois representations. This project is concerned with the following two objectives.1. Distinguishing Galois representations. The idea is to look for finer invariants of underlying objects that will enable us to conclude a contradiction and deduce that the original equation has no solutions.2. The Darmon programme. Henri Darmon proposed a far reaching generalization of Frey elliptic curves. He associates to certain ternary Diophantine problems hypergeometric abelian varieties. This objective is concerned with making the Darmon programme sufficiently practical to enable the resolution of particular Diophantine equations. Work performed from the beginning of the project to the end of the period covered by the report and main results achieved so far Work package 1.1. This has now been completed, and has resulted in four papers by the researcher in collaboration with Alain Kraus (Paris), and Samir Siksek (Warwick).Work package 1.2. The work on this package has been carried out by the researcher in collaboration with Lassina Dembele (Dartmouth) and John Voight (Dartmouth). The package is now in an advanced stage of completion, with one paper describing the theoretical aspects of the work complete, and another algorithmic paper in preparation.Work packages 2.1 and 2.2. Due to the 18 months that elapsed between the original proposal submission and the start of the project, much of the work for these work packages was carried out before the project started. This has resulted in a paper of the researcher in collaboration with Nicolas Billerey (Clermon-Ferrand), Imin Chen (Vancouver) and Luis Dieulefait (Barcelona). It became clear during that period that work on these packages should be combined, and that it should focus more closely on the Darmon programme than originally envisaged. The collaboration has continued during the project, and other papers are in preparation. Progress beyond the state of the art and expected potential impact (including the socio-economic impact and the wider societal implications of the project so far) The work on the project resulted in striking applications to Fermat's Last Theorem over number fields. For example, the researcher along with Kraus and Siksek, proved the asymptotic Fermat's Last Theorem over the layers of the Z_2 extension of the rationals. This is the first time that the Fermat equation has been resolved for an infinite family of number fields of arbitrarily large dimension.The work has also resulted in the most dramatic successes to date of the Darmon programme for Fermat equations of signature (r,r,p). Word cloud for the paper "Class Field Theory, Diophantine Analysis and Asymptotic Fermat"
