Asymptotic Fermat for signatures (p,p,2)$(p,p,2)$ and (p,p,3)$(p,p,3)$ over totally real fields

Abstract

Let be a totally real number field and consider a Fermat-type equation + = over . We call the triple of exponents (, , ) the signature of the equation. We prove various results concerning the solutions to the Fermat equation with signature (, , 2) and (, , 3) using a method involving modularity, level lowering and image of inertia comparison. These generalize and extend the recent work of Iik, Kara and zman. For example, consider a totally real field of degree with 2 + and 2 inert. Moreover, suppose there is a prime 5 which totally ramifies in and satisfies gcd(, -1) = 1, then we know that the equation + = 2 has no primitive, non-trivial solutions (, , ) 3 with 2| for sufficiently large.

Topics & Concepts