STNB 2014 (28è any)

The Fermat Equation over totally real fields


Nuno Freitas


Jarvis and Meekin showed that the strategy leading to the proof ofFermat's last theorem generalizes to proving that the classical Fermat equation x^p + y^p = z^p has no non-trivial solutions over  Q(\sqrt{2}).  

Two of the major obstacles to extending this result to other number fields are the modularity of the Frey curves and the existence of newforms in the spaces obtained after level lowering.

In this talk, we will discuss how recent modularity theorems, along with applying level lowering twice, allow us to circumvent the aforementioned obstacles. In particular, for infinitely many real quadratic fields K, we will show that there is a constant B_K (depending only on K) such that, for primes p > B_K, the equation x^p + y^p = z^p only admits solutions (a,b,c) satisfying abc=0.


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