On Fermat's equation over some quadratic imaginary number fields (Q2316174)

The furious activity following Wiles's proof of the classical case of Fermat's Last Theorem led to a proof of the FLT over all real quadratic fields. Freitas, Le Hung and Siksek proved the modularity of elliptic curves over real quadratic fields which led to the proof of an asymptotic version of FLT for at least five-sixth of these fields. Sengun and Siksek assumed the truth of two deep conjectures in the Langlands Program to prove an asymptotic version of FLT for general number fields \(K\) in the sense that there is a constant \(B_K\) so that for each prime \(p>B_K\), the Fermat equation of degree \(p\) has only trivial solutions in \(K\). Over imaginary quadratic fields, modularity of elliptic curves is not known and the units in the ring of integers are very few. There are also other issues in formulating and using an analogue of Serre's conjecture on Galois representations of imaginary quadratic fields, such as the fact that all mod-\(p\) eigenforms are not reductions of complex eigenforms. In their work, Sengun and Siksek circumvent these problems by going to large enough primes but their results are no longer effective versions. In the present paper, the author basically uses the technique of Sengun and Siksek and treates the three fields \(\mathbb{Q}(\sqrt{d})\) for \(d=-1,-2,-7\). Essentially, the author deals with imaginary quadratic fields of class number \(1\) but the above three fields among them are small enough to manage computations in the cohomology groups of the corresponding locally symmetric spaces. The author hopes to extend his results to all imaginary quadratic fields of class number \(1\). More precisely, the author assumes the truth of the following analogue of Serre's conjecture on Galois representations of the Galois group over \(\mathbb{Q}\) (now a theorem of Khare and Wintenberger) for the three fields \(K = \mathbb{Q}(\sqrt{d})\) for \(d=-1,-2,-7\). Conjecture. Let \(\overline{\rho} : G_K \rightarrow GL_2(\overline{\mathbb{F}_p})\) be an irreducible, continuous representation with prime-to-\(p\) part \(N\) of the Artin conductor and trivial prime-to-\(p\) character of \(det(\overline{\rho})\). Suppose that \(p\) is unramified in \(K\) and that \(\overline{\rho}|_{G_{K_P}}\) arises from a finite-flat group scheme over \(O_{K_P}\) for every \(P\) lying over \(p\). Then, there exists a mod-\(p\) eigenform \(\Phi : T_{\mathbb{F}_p}(Y_0(N)) \rightarrow \overline{\mathbb{F}_p}\) such that for all primes \(\pi\) coprime to \(pN\) in \(O_K\), we have \(Tr (\overline{\rho}(\mathrm{Frob}_{(\pi)}) = \Phi(T_{\pi})\). Here, \(T_{\mathbb{F}_p}(N)\) is the commutative Hecke algebra generated by Hecke operators \(T_{\pi}\) associated to the primes \(\pi\) coprime to \(pN\). The main theorem of the paper asserts: Theorem. Let \(K = \mathbb{Q}(\sqrt{d})\) for \(d=-1,-2\) or \(-7\). Assume that \(K\) satisfies the above conjectural version of Serre's conjecture. If \(p>3\) is prime, then \(a^p+b^p+c^p-0\) has no solutions for \(a,b,c \in K^{\ast}\). As a corollary, the author can deduce that the Fermat equation of any degree \(n \geq 3\) does not have any nontrivial solutions in \(\mathbb{Q}(i)\). In this deduction, the other two fields could not be included because of two reasons. Over \(\mathbb{Q}(\sqrt{-7})\), one has nontrivial solutions to \(a^4+b^4=c^4\) like \((a,b,c) = (1- \sqrt{-7}, 1+ \sqrt{-7},2)\). Over \(\mathbb{Q}(\sqrt{-2})\), the cubic equation \(a^3+b^3+c^3=0\) has infinitely many nontrivial solutions because this cubic curve is an elliptic curve of rank \(1\) over this field.