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

Abstract

Let K be a totally real number field and consider a Fermat‐type equation Aap+Bbq=Ccr$Aa^p+Bb^q=Cc^r$ over K. We call the triple of exponents false(p,q,rfalse)$(p,q,r)$ the signature of the equation. We prove various results concerning the solutions to the Fermat equation with signature false(p,p,2false)$(p,p,2)$ and false(p,p,3false)$(p,p,3)$ using a method involving modularity, level lowering and image of inertia comparison. These generalize and extend the recent work of Işik, Kara and Özman. For example, consider K a totally real field of degree n with 2∤hK+$2 \nmid h_K^+$ and 2 inert. Moreover, suppose there is a prime q⩾5$q\geqslant 5$ which totally ramifies in K and satisfies trueprefixgcdfalse(n,q−1false)=1$\gcd (n,q-1)=1$, then we know that the equation ap+bp=c2$a^p+b^p=c^2$ has no primitive, non‐trivial solutions (a,b,c)∈OK3$(a,b,c) \in \mathcal {O}_K^3$ with 2false|b$2 | b$ for p sufficiently large.

On ternary Diophantine equations of signature(p,p,2)over number fields

“…In [IKÖ20,IKÖ22], Işık, Kara and Özman studied the solutions of the Fermat-type equations x p + y p = z 2 and x p + y p = z 3 over various number fields utilising a similar approach carried out in [FS15a,ŞS18]. In [Moc22], Mocanu improved the results in [IKÖ20,IKÖ22] and proved similar results over totally real number fields. Following the ideas in [Fre15] to construct the relevant Frey curve, Mocanu proved asymptotic results for the Fermat-equation x p + y p = z r over totally real number fields in [Moc23].…”

On ternary Diophantine equations of signature over number fields

“…We have summarized these results in [14]; therefore, we will not repeat them here, but we want to mention that not many results exist for generalizations of these to higher degree number fields. During the write-up of this paper, we have been informed about the work of Mocanu [23] where she improves the results in [14] and proves similar versions for the Diophantine equation of signature (p, p, 3). In this paper, we also study the solutions of x p + y p = z 3 over number fields.…”