Rigid local systems, Hilbert modular forms, and Fermat's last theorem
This paper gives a very nice overview of a strategy for proving the non-existence of nontrivial primitive integral solutions to the generalized Fermat equation \(x^p + y^q = z^r\) for certain exponent triples \((p,q,r)\), along the lines of the proof of Fermat's last theorem (FLT). The overall strategy is explained in the introduction. 1. To a putative solution \((a,b,c)\), associate a 2-dimensional mod~\(p\) Galois representation \(\rho\) over a suitable totally real number field \(K\) with ramification bounded independently of \((a,b,c)\). In the FLT case, this is provided by the \(p\)-torsion of the associated Frey elliptic curve. 2. Show that \(\rho\) is modular, i.e., it comes from a Hilbert modular form over \(K\). This was done by Wiles in the FLT case. 3. Deduce that \(\rho\) comes from a modular form of small level, and thence that its image is small (in favorable cases). This is Ribet's level-lowering result. 4. Derive a contradiction by showing that the image of \(\rho\) has to be large. In the FLT case, this is provided by Mazur's results on rational isogenies of elliptic curves. These steps are then discussed in detail in the following sections. The first of these shows which Galois representations one can choose and identifies them as coming from torsion points on certain abelian varieties. The next section discusses modularity. Modularity of the relevant Galois representations would follow from the general modularity conjecture for abelian varieties of GL\(_2\)-type. In certain cases, the abelian varieties are elliptic curves, and so their modularity is known, see \textit{C. Breuil, B. Conrad, F. Diamond} and \textit{R. Taylor} [J. Am. Math. Soc. 14, 843-939 (2001; Zbl 0982.11033)]. This leads to a proof that no nontrivial primitive solutions exist for the equations \(x^p + y^p = z^2\) (\(p \geq 4\)) and \(x^p + y^p = z^3\) (\(p \geq 3\)), see \textit{H. Darmon} [Int. Math. Res. Not. 1993, 263-274 (1993; Zbl 0805.11028)] and \textit{H. Darmon} and \textit{L. Merel} [J. Reine Angew. Math. 490, 81-100 (1997; Zbl 0976.11017)]. In general, one can show that some other Galois representation on torsion points is modular, and so the modularity of \(\rho\) would follow from a generalization of Wiles' lifting result. The final two sections discuss the level-lowering and to which extent \(\rho\) can be shown to have small image, and (conjectural) Galois theoretic properties of torsion points that would force \(\rho\) to have large image.
scientific article; zbMATH DE number 1455496
scientific article