On ternary Diophantine equations of signature over number fields

Abstract

In this paper, we prove results about solutions of the Diophantine equation + = 3 over various number fields using the modular method. Firstly, by assuming some standard modularity conjecture we prove an asymptotic result for general number fields of narrow class number one satisfying some technical conditions. Secondly, we show that there is an explicit bound such that the equation + = 3 does not have a particular type of solution over = Q( √ − ) where = 1, 7, 19, 43, 67 whenever is bigger than this bound. During the course of the proof we prove various results about the irreducibility of Galois representations, image of inertia groups and Bianchi newforms.

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

“…More recently, Işik, Kara and Özman proved in [15] a similar asymptotic result for signature (𝑝, 𝑝, 3) over general number fields 𝐾 with narrow class number one satisfying some technical conditions. In the Appendix, they show how this result can be adapted to signature (𝑝, 𝑝, 2).…”

“…Let 𝔓 ∈ 𝑆 𝐾 and (𝑎, 𝑏, 𝑐) with 𝔓|𝑏 and prime exponent 𝑝 > 3𝑣 𝔓 (3). Let 𝐸 be the Frey curve in (14) with 𝑗-invariant 𝑗 𝐸 . Then 𝐸 has potentially multiplicative reduction at 𝔓 and 𝑝|#𝜌 𝐸,𝑝 (𝐼 𝔓 ).…”

“…Suppose (𝑎, 𝑏, 𝑐) ∈  3 𝐾 is a non-trivial, primitive solution to 𝑎 𝑝 + 𝑏 𝑝 = 𝑐 3 with prime exponent 𝑝 > 𝐵 𝐾 such that P|𝑏. Write 𝐸 for the Frey curve (14). Then, there is an elliptic curve 𝐸 ′ over 𝐾 such that:…”

“…We now present a result by Işik, Kara and Özman, proved in [14] which serves as the starting point of this paper. It gives a computable criteria of testing if the AFLT holds for certain type of solutions of the equations with signatures (𝑝, 𝑝, 2).…”

“…In [14], Işik, Kara and Özman list all known cases where Equation (1) has been solved over the rational integers in two tables (p. 4). Table 1 contains all unconditional results for infinitely many primes.…”