Barcelona Fall Workshop on Number Theory

On the generalized Fermat equation $x^{2\ell}+y^{2m}=z^p$


Samuele Anni


In this talk I will show that the generalized Fermat equation $x^{2\ell}+y^{2 m}=z^p$ has no non-trivial primitive solutions for primes $\ell, m \ge 5$, and $3 \le p \le 13$. This is achieved by relating a putative solution to a Frey curve over a real subfield of the $p$-th cyclotomic field, and studying its mod $\ell$ representation using modularity and level lowering. In particular, I will describe, on the one hand, the modularity theorem  for semistable elliptic curves over totally real number field used and, on the other hand, the computation with Hilbert modular forms done. This is joint work with Samir Siksek.


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