Abstract
Assuming a deep but standard conjecture in the Langlands programme, we prove Fermat’s Last Theorem over {mathbb {Q}}(i). Under the same assumption, we also prove that, for all prime exponents p ge 5, Fermat’s equation a^p+b^p+c^p=0 does not have non-trivial solutions over {mathbb {Q}}(sqrt{-2}) and {mathbb {Q}}(sqrt{-7}).
Highlights
Wiles’ remarkable proof of Fermat’s Last Theorem inspired mathematicians to attack the Fermat equation over number fields via elliptic curves and modularity
Modularity of elliptic curves over real quadratic fields was proved by Freitas et al [13]
Remark 2.2 When stated for more general number fields, this conjecture restricts to odd representations
Summary
Wiles’ remarkable proof of Fermat’s Last Theorem inspired mathematicians to attack the Fermat equation over number fields via elliptic curves and modularity. Sengün and Siksek [27] proved an asymptotic version of Fermat’s Last Theorem, under the assumption of two standard, but very deep conjectures in the Langlands programme. They prove that for a number field K, satisfying some special properties, there exists a constant BK , depending only on the field K, such that for all primes p > BK , the equation ap + bp + cp = 0, does not have solutions in K \{0}. Q(i), Q( −2) and Q( −7), we were able to make their result effective and to obtain optimal bounds for the constant BK Combining this with previous work on the Fermat equation over number fields, was sufficient to derive Theorem 1.1 below. Sengün and Siksek [27, Conjecture 4.1] is known to hold for base-change newforms via the theory of base change functoriality, we obtain a result that is dependant only on Serre’s modularity conjecture (see Conjecture 2.1)
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