Periodic Reporting for period 1 - GalRepsDiophantine (Galois Representations and Diophantine Problems)
Okres sprawozdawczy: 2018-03-01 do 2020-02-29
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 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.
The work has also resulted in the most dramatic successes to date of the Darmon programme for Fermat equations of signature (r,r,p).