On the solutions of $x^p+y^p=2^r z^p$, $x^p+y^p=z^2$ over totally real fields

In this article, we study the non-trivial primitive solutions of a certain type for the Diophantine equations $x^p+y^p=2^rz^p$ and $x^p+y^p=z^2$ of prime exponent $p$, $r \in \mathbb{N}$, over a totally real field $K$. Then for $r=2,3$, we study the non-trivial primitive solutions over $\mathcal{O}_K$ for the equation $x^p+y^p=2^rz^p$ of prime exponent $p$. Finally, we give several purely local criteria for $K$ such that the equation $x^p+y^p=2^rz^p$ has no non-trivial primitive solutions over $\mathcal{O}_K$.


Introduction
Throughout this article, K denotes a totally real number field. Let P , P denote Spec(O K ), Spec(Z), respectively. In the literature, the thrust to understand the Diophantine equations, especially the Fermat type equations over number fields, has a long and enthralling history. Though there were many approaches available, the cyclotomic and Mordell-Weil methods were proved to be more successful to understand these equations. In fact, the modularity of elliptic curves was crucial in the proof of Fermat's Last Theorem by Wiles (cf. [Wil95, Theorem 0.5]).
In [JM04, Theorem 1.3], Jarvis and Meekin show that the Fermat equation x n + y n = z n of exponent n ∈ N has no non-trivial solutions over Z[ √ 2] for n ≥ 4. In [Tur18], [Tur20], Ţ urcaş studied the Fermat equation x p + y p = z p of exponent p ∈ P, over imaginary quadratic fields of class number 1.
In [FS15a, Theorem 3], Freitas and Siksek show that the asymptotic FLT holds for the Fermat equation x p + y p = z p over K, i.e., there exists a constant B K > 0 (depends on K) such that for primes p > B K , the equation x p +y p = z p of exponent p has no non-trivial solutions over K. The proof depends upon certain explicit bounds on the solutions of S-unit equation (3.2). In [SS18, Theorem 1.1], Şengün and Siksek, extending the work in [FS15a], show that the asymptotic FLT holds for the Fermat equation x p +y p = z p over number fields. In [Dec16, Theorem 1], Deconinck extended the work in [FS15a] to the Fermat equation Ax p + By p = Cz p with 2 ∤ ABC. Later, in [KO20], Kara and Ozman extended the work in [Dec16] to number fields.
In this article, we study the non-trivial primitive solutions (over K) of the Diophantine equation x p + y p = 2 r z p with p ∈ P, r ∈ N. Note that this equation is complementary to the one considered by Deconinck. We also study a similar question for the equation x p + y p = z 2 of exponent p ∈ P over K.
B K ) depending on K, r (resp., on K) such that for primes p > B K,r (resp., B K ), the equation x p + y p = 2 r z p (resp., x p + y p = z 2 ) of exponent p has no non-trivial primitive solutions in S.
In [Rib97, Theorem 3], Ribet shows that there are no non-zero integer solutions to the equation x p + 2 r y p + z p = 0 of exponent p ∈ P with 1 ≤ r < p. In [DM97], Darmon and Merel show that the equation x n + y n = 2z n of exponent n ∈ N has no non-trivial primitive integer solutions for n ≥ 3.
In this article, we first show that the equation x p + y p = 2 r z p of exponent p ∈ P, with r ∈ N, has no asymptotic solution in W K , where W K is as in Definition 3.2 (cf. Theorem 3.3). Later, for r = 2, 3, we show that the equation x p + y p = 2 r z p of exponent p ∈ P has no asymptotic solution in O 3 K (cf. Theorem 3.4). The proofs of Theorem 3.3 and Theorem 3.4 depend on certain explicit bounds on the solutions of S K -unit equation (3.2).
In [DM97], Darmon and Merel show that the equation x n + y n = z 2 of exponent n ∈ N has no non-trivial primitive integer solutions for n ≥ 4. In [IKO20, Theorem 1.1], Işik, Kara and Ozman show that, for any totally real field K with narrow class number h + K = 1 and if there exists P ∈ P over 2 with residual degree 1, then the equation x p + y p = z 2 of exponent p ∈ P has no asymptotic solution of certain type over K. This proof depends upon the certain explicit bounds on the solutions of the S K -unit equation (5.2). In [IKO22, Theorem 6.1], they extended this work to number fields.
In this article, we relax the assumption on the existence of P ∈ P over 2 with residual degree 1 and h + K = 1 in [IKO20, Theorem 1.1](cf. Theorem 5.3). More precisely, we show that the equation x p +y p = z 2 of exponent p ∈ P has no asymptotic solution in W ′ K (cf. Definition 5.2 for W ′ K ). This proof depends upon certain explicit bounds on the solutions of the S K -unit equation (5.2). Recently, Mocanu generalized [IKO20, Theorem 1.1], but with a different hypothesis (cf. [Moc22, Theorem 3]). Namely, from the assumption h + K = 1 to Cl S K (K) = 1. In this case, the proof depends upon some explicit bounds on the solutions of α + β = γ 2 , with α, β ∈ O * S K and γ ∈ O S K (cf. [Moc22, page 3] for the definition of Cl S K (K), O S K and O * S K ). In [FKS21], the authors gave some purely local criteria for K such that the asymptotic FLT holds for the Fermat equation x p + y p = z p over K. In the last section, we give several purely local criteria for K such that the equation x p + y p = 2 r z p (resp., x p + y p = 2 r z p with r = 2, 3) of exponent p ∈ P has no asymptotic solution in W K (resp., in O 3 K ). In the proofs of Theorems 3.3, 3.4 and 5.3, we use the modularity of the Frey curve E attached to a non-trivial primitive solution, irreducibility of the residual Galois representationρ E,p , semi-stable reduction of E at odd primes and level lowering.
1.1. Structure of the article: The article is organized as follows. In §2, we collate all the preliminaries required to prove main results. In §3 and §4 (resp., §5 and §6), we state and prove Theorem 3.3, 3.4 (resp., Theorem 5.3) for the equation x p +y p = 2 r z p (resp., x p + y p = z 2 ) of exponent p ∈ P.

Preliminaries
In this section, we recall some preliminaries and known results required to prove the main theorems of this article. Throughout this article, we denote O K , n and p for the ring of integers of K, an ideal of O K and a rational prime, respectively.
Theorem 2.1. Up to isomorphism overK, there are only a finite number of elliptic curves E over K which are not modular. Further, if K is real quadratic, then every elliptic curve over K is modular.
2.2. Eichler-Shimura. We now state a standard conjecture for K, which is a generalization of the Eichler-Shimura theorem over Q.
Conjecture 2.2 (Eichler-Shimura). Let f be a Hilbert modular newform over K of parallel weight 2, level n and with rational eigenvalues. Then there exists an elliptic curve E f /K of conductor n having same L-function as f .
In [Bla04], Blasius based on the work of Hida [Hid81], gave a partial answer to Conjecture 2.2. More precisely: Theorem 2.3. Let f be a Hilbert modular newform over K of parallel weight 2, level n and with rational eigenvalues. Suppose that either [K : Q] is odd or there is a finite prime q of K such that v q (n) = 1. Then there exists an elliptic curve E f /K of conductor n having same L-function as f .
For any elliptic curve E/K, letρ E,p : G K := Gal(K/K) → Aut(E[p]) ≃ GL 2 (F p ) be the residual Galois representation of G K , acting on the p-torsion points of E. For any Hilbert modular newform f over K of weight k, level n and character Ψ with coefficient field Q f , letρ f,λ : G K → GL 2 (F λ ) be the residual Galois representation attached to f, λ, where λ ∈ Spec(O Q f ).
In [FS15a, Corollary 2.2], Freitas and Siksek provided a partial answer to Conjecture 2.2 in terms of mod p Galois representations. More precisely: Theorem 2.4. Let E be an elliptic curve over K and p be an odd prime. Suppose thatρ E,p is irreducible andρ E,p ∼ρ f,p for some Hilbert modular newform f over K of parallel weight 2, level n with rational eigenvalues. Let q ∤ p be a prime of K such that (1) E has potentially multiplicative reduction at q, (2) p|#ρ E,p (I q ) and p ∤ (Norm(K/Q)(q) ± 1). Then there exists an elliptic curve E f /K of conductor n having same L-function as of f .

2.3.
Irreducibility of mod p representations of elliptic curves. In [FS15b, Theorem 2], Freitas and Siksek gave a criterion for determining the irreducibility of ρ E,p , for any elliptic curve E over K. More precisely: Theorem 2.5. Let K be a totally real Galois field. Then there exists an effective constant C K (depends on K) such that, if p > C K is a prime and E/K is an elliptic curve over K which is semi-stable at all q|p, thenρ E,p is irreducible.
2.4. Level lowering. Let E/K be an elliptic curve of conductor n. For a prime ideal q of K, let ∆ q be the discriminant of a minimal local model of E at q. Let m p := p|vq(∆q), q||n q and n p := n m p . (2.1) In [FS15a,Theorem 7], the authors talked about the level lowering of mod p Galois representations attached to elliptic curves over K. More precisely, they prove Theorem 2.6. Let E/K be an elliptic curve of conductor n. Let p be a rational prime. Suppose that the following conditions hold: (1) For p ≥ 5, the ramification index e(q/p) < p − 1 for all q|p and Q(ζ p ) + K; (2) E/K is modular andρ E,p is irreducible; (3) E is semi-stable at all q|p and p|v q (∆ q ) for all q|p. Then there exists a Hilbert modular newform f of parallel weight 2, level n p and some prime λ of Q f such that λ|p andρ E,p ∼ρ f,λ .
3. Solutions of the Diophantine equations x p + y p = 2 r z p over totally real fields K In this section, we study the solutions of the following equation over K.
x p + y p = 2 r z p (3.1) with prime exponent p > 5 and r ∈ N.
For any S ⊆ P := Spec(O K ), let O S := {α ∈ K : v P (α) ≥ 0 for all P ∈ P \ S} denote the ring of S-integers in K and O * S denote the units of O S . We refer to them as S-units. Let S K := {P ∈ P : P|2} and U K := {P ∈ S K : (3, v P (2)) = 1}.
Definition 3.2. Let W K be the set of (a, b, c) ∈ O 3 K satisfying the equation (3.1) of exponent p with P|abc for every P ∈ S K .
3.1. Main results. We now state the main results of this article.
Theorem 3.3. Let K be a totally real field. Suppose, for every solution (λ, µ) to the S K -unit equation there exists some P ∈ S K that satisfies Then the equation (3.1) of exponent p has no asymptotic solution in W K , i.e., there exists a constant B K,r (depends on K, r) such that for primes p > B K,r , the equation (3.1) of exponent p has no non-trivial primitive solutions in W K .
We write (ES) for "either [K : Q] ≡ 1 (mod 2) or Conjecture 2.2 holds for K." Theorem 3.4. Let K be a totally real field satisfying the condition (ES). Suppose, for every solution (λ, µ) to the S K -unit equation (3.2) there exists some P ∈ U K that satisfies Then the equation (3.1) of exponent p with r = 2, 3 has no asymptotic solution in O 3 K , i.e., there exists a constant B K (depends on K) such that for primes p > B K , the equation (3.1) of exponent p with r = 2, 3 has no non-trivial primitive solutions in O 3 K . Corollary 3.5. Let K and r be as in Theorem 3.4. Then the conclusion of Theorem 3.4 remains valid even if we replace both the assumptions in (3.4) by (3.5) Proof of Corollary 3.5. Let P ∈ U K and t := max {|v P (λ)|, |v P (µ)|} > 0. Since . Now, the proof follows from Theorem 3.4.

4.
Steps to prove Theorem 3.3 and Theorem 3.4 For any non-trivial primitive solution (a, b, c) ∈ O 3 K to the equation (3.1) of exponent p > 5, consider the Frey curve 4.1. Modularity of the Frey curve. We will now prove the modularity of the Frey curve E given by (4.1) over K.
K be a non-trivial primitive solution to the equation (3.1) of exponent p. Let E/K be the Frey curve attached to (a, b, c)(cf. (4.1)). Then there exists a constant A K,r (depends on K, r) such that for primes p > A K,r , E/K is modular.
By Theorem 2.1, there exists finitely manyK-isomorphism classes of non-modular elliptic curves over K. Let j 1 , ..., j t ∈ K be the j-invariants of those elliptic curves.
For each i = 1, 2, . . . , t, the equation j(λ) = j i has at most six solutions in K.
The above two equations determine p uniquely and denote it by p k . Suppose p = q are primes such that b a p = b a q , which means b a is a root of unity. Similarly, c a is also a root of unity. We get b = ±a, c = ±a, as K is totally real. Since (a, b, c) ∈ O 3 K is primitive, this gives a = ±1, b = ±1, c = ±1 and hence the solution (a, b, c) is trivial. The proof of Theorem 4.1 follows by taking A K,r = max{p 1 , ..., p m }.

4.2.
Reduction type. In this section, we describe the reduction of the Frey curve E := E a,b,c attached to a non-trivial primitive solution (a, b, c) to the equation (3.1) at q ∈ P . 4.2.1. Reduction type at odd primes. The following lemma characterizes the type of reduction of the Frey curve E at primes q away from S K .
K be a non-trivial primitive solution to the equation (3.1) of exponent p. Let E := E a,b,c be the associated Frey curve as in (4.1). Then at all primes q ∈ P away from S K , E is minimal, semi-stable and satisfies p|v q (∆ E ). Let n be the conductor of E and n p be as in (2.1). Then is primitive solution to the equation (3.1) and q ∤ 2, q divides precisely one of a, b and c. Recall that c 4 = 2 4 (a 2p + 2 r b p c p ), so v q (c 4 ) = 0. Hence, E is minimal and has multiplicative reduction at q. Therefore, E is semi-stable at q. Since v q (∆ E ) = 2p (v q (abc)), p|v q (∆ E ). By (2.1), we get q ∤ n p for all q ∈ P \ S K . For P ∈ S K , r P = v P (n) ≤ 2 + 6v P (2) (cf. Lemma 4.3. Let E/K be an elliptic curve. Let p > 5 be a prime and q ∤ p be a prime of K. Then E has potentially multiplicative reduction at q i.e., v q (j E ) < 0 and p ∤ v q (j E ) if and only if p|#ρ E,p (I q ).
The next lemma specifies the type of reduction of the Frey curve E at primes q ∤ 2.
K be a non-trivial primitive solution to the equation (3.1) of exponent p > 5 and let E := E a,b,c be the associated Frey curve as in (4.1). For q ∈ P , if q ∤ 2p then p ∤ #ρ E,p (I q ).
Proof. By Lemma 4.3, it is enough to show that either v q (j E ) ≥ 0 or p|v q (j E ). If q ∤ ∆ E , then E has good reduction at q, hence v q (j E ) ≥ 0. If q|∆ E then q|abc. Since (a, b, c) is a primitive solution to the equation (3.1) and q ∤ 2, q divides exactly one of a, b and c. Therefore, v q (c 4 ) = 0, v q (j E ) = −2p(v q (abc)) and hence we are done. 4.2.3. Reduction type at primes P ∈ S K . The following lemma is quite useful.
K be a non-trivial primitive solution to the equation (3.1) of exponent p > [K : Q]r. For P ∈ S K , if P|abc, then P|c. In particular, P ∤ ab.
Proof. If P ∤ c, then P|ab. So, either P|a or P|b. Since P|2 and a p + b p = 2 r c p with r ∈ N, P divides both a and b, which implies P p |2 r c p . This cannot happen by the unique factorization of ideals in O K and p > [K : Q]r.
We now recall [FS15a, Lemma 3.6], which is useful for determining the type of reduction of the Frey curve E at primes P ∈ U K .
Lemma 4.6. Let E/K be an elliptic curve. Let p ≥ 3 be a prime and P ∈ S K . Suppose E has a potential good reduction at P. Then 3 ∤ v P (∆ E ) if and only if 3|#ρ E,p (I P ).
We will now discuss the type of reduction of the Frey curve E := E a,b,c at P ∈ S K for (a, b, c) ∈ W K and at P ∈ U K for (a, b, c) ∈ O 3 K . More precisely: K is a non-trivial primitive solution to the equation (3.1) of exponent p > max {|(4 − r)v P (2)|, [K : Q]r} and let E := E a,b,c be the associated Frey curve as in (4.1).

Proofs of Theorem 3.3 and Theorem 3.4.
4.3.1. Proof of Theorem 3.3. The proof of this theorem depends on the following auxiliary result.
Theorem 4.8. Let K be a totally real field. Then there is a constant B K,r > 0 (depends on K, r) such that the following hold. Let (a, b, c) ∈ W K be a non-trivial primitive solution to the equation (3.1) of exponent p > B K,r and let E := E a,b,c be the associated Frey curve as in (4.1). Then there exists an elliptic curve E ′ /K such that: (1) E ′ /K has good reduction away from S K and has full 2-torsion, i.e., |E ′ (K)[2]| = 4; (2)ρ E,p ∼ρ E ′ ,p and v P (j E ′ ) < 0 for all P ∈ S K . Proof of Theorem 4.8. By Lemma 4.1, E is modular for all primes p > A K,r . By Lemma 4.2, the Frey curve E is semi-stable away from S K . If necessary, we can take the Galois closure of K to ensure thatρ E,p is irreducible for all primes p ≫ 0 (cf. Theorem 2.5).
By Theorem 2.6, there exist a Hilbert modular newform f of parallel weight 2, level n p and some prime λ of Q f such that λ|p andρ E,p ∼ρ f,λ for p ≫ 0. By allowing p to be sufficiently large, we can assume Q f = Q (cf. [FS15a, §4] for more details).
Let P ∈ S K . Then E has potential multiplicative reduction at P and p|#ρ E,p (I P ) for p > max {|(4 − r)v P (2)|, [K : Q]r} (cf. Lemma 4.7). The existence of an elliptic curve E f of conductor n p then follows from Theorem 2.4 for all p ≫ 0 (also excluding the primes p | (Norm(K/Q)(P) ± 1)). Therefore,ρ E,p ∼ρ E f ,p for all primes p > B K,r , where B K,r is the maximum of all the above implicit lower bounds.
(1) Since the conductor of E f is n p given in (4.2), E f has good reduction away from S K . After enlarging B K,r by a sufficient amount and by possibly replacing E f with an isogenous curve, say E ′ , we will find that E ′ /K has full 2-torsion. This follows from [Coh07, Proposition 15.4.2] and the fact that E/K has all its points of order 2 (cf. [FS15a,§4] for more details). The elliptic curve E ′ has good reduction away from S K .
We now prove Theorem 3.3.
Theorem 4.9. Let K be a totally real field satisfying (ES). Then there exists a constant B K (depends on K) such that the following hold. Let (a, b, c K be a non-trivial primitive solution to the equation (3.1) of exponent p > B K with r = 2, 3 and let E := E a,b,c be the associated Frey curve as in (4.1). Then there exists an elliptic curve E ′ /K such that: (1) E ′ /K has good reduction away from S K and has full 2-torsion, i.e., |E ′ (K)[2]| = 4 andρ E,p ∼ρ E ′ ,p ; Proof. Most of the proof is similar to that of Theorem 4.8 except for the last part. Suppose P ∈ U K and r = 2, 3.
5. Solutions of the Diophantine equation x p + y p = z 2 over K In this section, we study the solutions of the following equation over K.
Definition 5.1 (Trivial solution). We say a solution (a, b, c) ∈ O 3 K to the equation (5.1) of exponent p, is trivial if abc = 0, otherwise non-trivial. Further, we call it as primitive if a, b, c are pairwise co-prime.
Definition 5.2. Let W ′ K be the set of (a, b, c) ∈ O 3 K satisfying the equation (5.1) of exponent p with P|ab for every P ∈ S K .

Main result.
For any positive integer m, we define the m-Selmer group of K and S K by By [Coh00, §5.2.2 and 7.4], K(S K , m) is a finite abelian group, for every K and m. Let L = K( √ a) for a ∈ K(S K , 2). Let S L be the set of all prime ideals of L lying over primes of S K . Consider S K -unit (resp., S L -unit) equation We now state the main result of this section.
Theorem 5.3. Let K be a totally real field. For each a ∈ K(S K , 2), let L = K( √ a).
Suppose, for every solution (λ, µ) to the S K -unit solution (5.2), there exists some P ∈ S K that satisfies and for every solution (λ, µ) to the S L -unit equation (5.2), there exists some P ′ ∈ S L that satisfies max {|v P ′ (λ)|, |v P ′ (µ)|} ≤ 4v P ′ (2). (5.4) Then the equation (5.1) of exponent p has no asymptotic solution in W ′ K , i.e., there exists a constant B K (depends on K) such that for primes p > B K , the equation (5.1) of exponent p has no non-trivial primitive solutions in W ′ K . The theorem above can be thought of as a generalization of [IKO20, Theorem 1.1], where they used the assumptions T K = ϕ and h + K = 1 to prove it. However, Theorem 5.3 is independent of these assumptions.

Steps to prove Theorem 5.3
For any non-trivial primitive solution (a, b, c) ∈ O 3 K to the equation (5.1) of exponent p, consider the Frey curve where c 4 = 2 6 (4c 2 − 3a p ) = 2 6 (a p + 4b p ), ∆ E = 2 12 (a 2 b) p and j E = 2 6 (a p +4b p ) 3 (a 2 b) p . 6.1. Modularity of the Frey curve. We now study the modularity of the Frey curve E given by (6.1) over K.
Theorem 6.1. Let (a, b, c) ∈ W ′ K be a non-trivial primitive solution to the equation (5.1) of exponent p and let E := E a,b,c be the associated Frey curve as in (6.1). Then there exists a constant A K (depends on K) such that for primes p > A K , E/K is modular.
Proof. The proof of this theorem is similar to the proof of Theorem 4.1, except that, here, j E = 2 6 (a p +4b p ) 3 (a 2 b) p , j(λ) = 2 6 (4λ−1) 3 λ for λ = − b p a p . So, there exists λ 1 , λ 2 , ..., λ m ∈ K such that E is modular for all λ / ∈ {λ 1 , λ 2 , ..., λ m }. If λ = λ k for some k ∈ {1, 2, . . . , m}, then b a p = −λ k . The above equation determines p uniquely and denotes it p k . If not, let p = q are primes such that b a p = b a q , which means b a is a root of unity. Since K is totally real, we get b = ±a. Again, since (a, b, c) is primitive, a = ±1 and b = ±1, which is not possible since (a, b, c) ∈ W ′ K . Hence the proof follows by taking A K = max{p 1 , ..., p m }.
6.2. Reduction type at odd primes. The following lemma is an analog of Işik, et al. (cf. [IKO20, Lemma 3.2]) and Lemma 4.2, which specifies the types of reduction of the Frey curve E given in (6.1) at primes q away from S K .
Lemma 6.2. Let (a, b, c) ∈ O 3 K be a non-trivial primitive solution to the equation (5.1) of exponent p and let E be the Frey curve in (6.1). Then at all primes q ∈ P away from S K , E is minimal, semi-stable and satisfies p|v q (∆ E ). Let n be the conductor of E and n p be as in (2.1). Then where 0 ≤ r ′ P ≤ r P ≤ 2 + 6v P (2). The following lemma describes the type of reduction of the Frey curve E = E a,b,c at P ∈ S K and (a, b, c) ∈ W ′ K . More precisely: Lemma 6.3. Suppose (a, b, c) ∈ W ′ K is a non-trivial primitive solution to the equation (5.1) of exponent p and let E := E a,b,c be the associated Frey curve as in (6.1). If P ∈ S K and p > 6v P (2), then p|#ρ E,p (I P ). The same conclusion also holds for P ∈ S L .
Proof. Suppose P ∈ S K and (a, b, c) ∈ W ′ K . By the definition of W ′ K , either P|a or P|b. Recall that j E = 2 6 (a p +4b p ) 3 (a 2 b) p . If P|a, then P ∤ b because (a, b, c) is primitive. So v P (j E ) = 6v P (2) + 3v P (a p + 4b p ) − 2pv P (a). Since p > 6v P (2), v P (j E ) = 12v P (2) − 2pv P (a) < 0 and p ∤ v P (j E ). The same argument holds even if P|b. The proof now follows from Lemma 4.3. 6.3. Proof of Theorem 5.3. The proof of this theorem depends on the following auxiliary result.. Theorem 6.4. Let K be a totally real field. Then there is a constant B K (depends on K) such that the following hold. Let (a, b, c) ∈ W ′ K be a non-trivial primitive solution to the equation (5.1) of exponent p > B K and let E := E a,b,c be the associated Frey curve as in (6.1). Then there exists an elliptic curve E ′ over K such that: (1) E ′ /K has good reduction away from S K and has a non-trivial 2-torsion point; (2)ρ E,p ∼ρ E ′ ,p and v P (j E ′ ) < 0 for P ∈ S K .
Proof. Arguing as in the proof of Theorem 4.8, there exists a constant B K (depending on K) and an elliptic curve E f /K of conductor n p such thatρ E,p ∼ρ E f ,p for all primes p > B K .
(1) Since the conductor of E f is n p given in (6.2), E f has good reduction away from S K . After enlarging B K by an efficient amount and by possibly replacing E f with an isogenous curve, say E ′ , we will find that E ′ /K has a non-trivial 2-torsion point (cf. [Moc22, page 1247] for more details). Also, E ′ has good reduction away from S K . (2) Since E f is isogenous to E ′ andρ E,p ∼ρ E f ,p , we haveρ E,p ∼ρ E ′ ,p . So by Lemma 6.3, we have p|#ρ E,p (I P ) = #ρ E ′ ,p (I P ) for any P ∈ S K . This gives v P (j E ′ ) < 0, by Lemma 4.3.
We will now give a proof of Theorem 5.3 and it is similar to that of [IKO20, Theorem 1.1].
Proof of Theorem 5.3. Let B K be as in Theorem 6.4 and let (a, b, c) ∈ W ′ K be a nontrivial primitive solution to the equation (5.1) of exponent p > B K . By Theorem 6.4, there exists an elliptic curve E ′ /K having a non-trivial 2-torsion point. Hence E ′ /K has a model of the form for some c, d ∈ K. Let L be the splitting field of x 3 + cx 2 + dx over K. Then either E ′ has full 2-torsion over K, or E ′ has full 2-torsion over L.
Suppose E ′ has full 2-torsion over K. Now, arguing as in the proof of Theorem 3.3 and by (5.3), we get v P (j E ′ ) ≥ 0 for some P ∈ S K , which is a contradiction to Theorem 6.4.
Suppose E ′ has full 2-torsion over L. By [Kou19, Proposition 3.1], L = K( √ a) for some a ∈ K(S K , 2). We now show that E ′ /L continues to have the same properties as in Theorem 6.4 over L. Namely; • Since E ′ has good reduction away from S K , E ′ /L has good reduction away from S L and has full 2-torsion over L. • By Theorem 6.4,ρ E,p ∼ρ E ′ ,p . By Lemma 6.3, we have p|#ρ E,p (I P ) = #ρ E ′ ,p (I P ) for any P ∈ S L . So by Lemma 4.3, v P (j E ′ ) < 0 for P ∈ S L .
Arguing as in the proof of Theorem 3.3 and by (5.4), we get v P (j E ′ ) ≥ 0 for some P ∈ S L . This is a contradiction to Theorem 6.4 over L.
7. Local criteria for the solutions of the Diophantine equation x p + y p = 2 r z p over K In this section, we give several purely local criteria for K where the equation (3.1) of exponent p has no asymptotic solution in O 3 K , in W K . First, we give local criteria for K of odd degree such that the equation (3.1) of exponent p with r = 2, 3 has no asymptotic solution in O 3 K . More specifically: Proposition 7.1. Let n = [K : Q] and l > 5 be a prime such that (n, l − 1) = 1. Suppose that l totally ramifies and 2 is inert in K. Then the equation (3.1) of exponent p with r = 2, 3 has no asymptotic solution in O 3 K . For example, one can take K = Q(α), where α satisfies the minimal polynomial x 3 − x 2 + 1. In K, 2 is inert, 23 is totally ramified in K and (3, 23 − 1) = 1.
Proposition 7.2. Let n = [K : Q] be odd and 3 ∤ n. Suppose 2 is inert and 3 totally splits in K. Then the conclusion of Proposition 7.1 holds.
We give local criteria for K with [K : Q] ≡ 1 (mod 2) such that the equation (3.1) of exponent p has no asymptotic solution in W K . These criteria are an analogue of [FKS21, Theorems 1,3] and their proofs are an application of Theorem 3.3.
Corollary 7.3. Suppose one of the hypothesis holds: (1) Suppose n = [K : Q] and l > 5 is a prime number such that (n, l − 1) = 1. Assume 2 is either inert or totally ramified in K and l totally ramifies in K.
(2) Suppose [K : Q] is odd, 2 is either inert or totally ramified in K and 3 totally splits in K. Then the equation (3.1) of exponent p has no asymptotic solution in W K .