Let \(K\) be a totally real number field of odd degree in which \(2\) is inert. Let \(l \geq 5\) be a prime with \(l \nmid
[K:\mathbb{Q}]\) and \(\gcd(\frac{l-1}{2}, [K:\mathbb{Q}])=1\). We prove that if \(l\) is non-Wieferich i.e., \(2^{l-1} \not\equiv 1 \pmod {l^2}\) and
\(l\) is totally ramified in \(K\), then the asymptotic Fermat’s Last Theorem holds over each \(n\)-th layer \(K_{n,l}\) of
the cyclotomic \(\mathbb{Z}_l\)-extension of \(K\). We then prove that the generalized Fermat equation \(Ax^p+By^p+Cz^p=0\) has no asymptotic solution over
each \(n\)-th layer \(K_{n,l}\), when \(A,B,C \in \{u2^r: u\in \mathcal{O}_K^\times, r \in \mathbb{Z}_{\geq 0}\}\). For any odd prime \(d\), we also prove that if \(A,B,C \in \{\pm 2^rd^s: r,s \in \mathbb{Z}_{\geq 0}\}\) and \(h_{\mathbb{Q}_{n,l}}^+\) is odd, then the generalized Fermat equation
\(Ax^p+By^p+Cz^p=0\) has no effective asymptotic solution \((a,b,c) \in \mathcal{O}_{\mathbb{Q}_{n,l}}^3\) with \(2|abc\). The effectivity in the case of
\(\mathbb{Q}_{n,l}\) follows from a result of Thorne, which proves the modularity of elliptic curves over \(\mathbb{Q}_{n,l}\).
Let \(K\) be a totally real number field and \(\mathcal{O}_K\) denote the ring of integers of \(K\). The generalized Fermat equation of exponent \(p\) over \(K\) is the equation \[\label{Ax94p43By94p43Cz94p610} Ax^p+By^p+Cz^p=0, \; A,B,C \in
\mathcal{O}_K\setminus \{0\}.\tag{1}\] The asymptotic Fermat’s Last Theorem over \(K\) is the statement that there is a bound \(V_K\) (depending only on \(K\)) such that for all primes \(p >V_K\), the only solutions to the equation \(x^p+y^p+z^p=0\) with \(x,y,z \in K\) are the
trivial ones, i.e., \(xyz=0\). If \(V_K\) is effectively computable, we shall refer to this as the effective asymptotic Fermat’s Last Theorem over \(K\).
In [1], Freitas and Siksek first proved that the asymptotic Fermat’s Last Theorem holds over \(K\) whenever \(K\) satisfy certain \(S\)-unit criterion, and proved that the effective asymptotic Fermat’s Last Theorem holds over \(K=\mathbb{Q}(\sqrt{d})\) for a subset of
\(d \geq 2\) with density \(\frac{5}{6}\) among the set of square-free integers \(d \geq 2\). In [2], Deconinck extended the work of [1] to 1 with \(ABC\)
is odd. In KS23?, Diophantine1?, Kumar and the author studied
the asymptotic solution of \(x^p+2^ry^p+z^p=0\) over \(K\), for positive integers \(r\). Finally in [3], the author extended the result of KS23?, Diophantine1? to 1 with \(ABC\) is even and computed the density of all square free integers \(d\geq
2\) such that the equation \(Ax^p+By^p+Cz^p=0\) has no effective asymptotic solution over \(K=\mathbb{Q}(\sqrt{d})\).
In [4], Freitas, Kraus and Siksek proved that the effective asymptotic Fermat’s Last Theorem holds over each \(n\)-th layer
\(\mathbb{Q}_{n,2}\) of the cyclotomic \(\mathbb{Z}_2\)-extension of \(\mathbb{Q}\) (cf. §2 for the \(n\)-th layer). For any totally real number field \(K\), in [5], the authors proved that if \(2\) is totally ramified in \(K\) and the narrow class number \(h_K^+\) of \(K\) is odd, then asymptotic Fermat’s Last Theorem
holds over any totally real \(2\)-extension of \(K\) unramified away from \(2\).
Let \(l\geq 5\) be a prime. In [6], the authors proved that if \(l\) is non-Wieferich,
i.e., \(2^{l-1} \not\equiv 1 \pmod {l^2}\), then the effective asymptotic Fermat’s Last Theorem holds over each \(n\)-th layer \(\mathbb{Q}_{n,l}\) of the
cyclotomic \(\mathbb{Z}_l\)-extension of \(\mathbb{Q}\). In this article, we study the asymptotic solutions of the generalized Fermat equation 1 over
the cyclotomic \(\mathbb{Z}_l\)-extensions of totally real number fields \(K\).
We first prove that the asymptotic Fermat’s Last Theorem holds over each \(n\)-th layer \(K_{n,l}\) of the cyclotomic \(\mathbb{Z}_l\)-extension of \(K\).
Theorem 1. Let \(K\) be a totally real number field of odd degree in which \(2\) is inert. Let \(l \geq 5\) be a prime with \(l \nmid [K:\mathbb{Q}]\) and \(\gcd(\frac{l-1}{2}, [K:\mathbb{Q}])=1\). Assume
\(l\) is non-Wieferich, i.e., \(2^{l-1} \not\equiv 1 \pmod {l^2}\);
\(l\) is totally ramified in \(K\).
Then the asymptotic Fermat’s Last Theorem holds over each \(n\)-th layer \(K_{n,l}\) of the cyclotomic \(\mathbb{Z}_l\)-extension of \(K\). Moreover, if all elliptic curves \(E/K_{n,l}\) with full \(2\)-torsion are modular, then the effective asymptotic Fermat’s Last Theorem holds over \(K_{n,l}\).
We now study the asymptotic solution of the generalized Fermat equation 1 over the cyclotomic \(\mathbb{Z}_l\) extension of \(K\).
Definition 2. Let \(K\) be a totally real field. A solution \((a, b, c)\in K^3\) to the equation@eq:Ax94p43By94p43Cz94p610 is said to be non-trivial if \(abc\neq 0\). We say a solution \((a, b, c)\in \mathcal{O}_K^3\) is primitive if \(a\mathcal{O}_K+b\mathcal{O}_K+c\mathcal{O}_K=\mathcal{O}_K\).
Definition 3. Let \(K\) be a totally real field and let \(S\) be a subset of \(\mathcal{O}_K^3\). We say the generalized Fermat
equation 1 has no asymptotic solution in \(K^3\) (resp. \(S\)), if there exists a constant \(V:=V_{K,A,B,C}\)
(depending on \(K,A,B,C\)) such that for primes \(p >V\), 1 has no non-trivial solutions in \(K^3\) (resp.
non-trivial primitive solutions in \(S\)). Moreover, if the above constant \(V\) is effectively computable, then we say 1 has no effective asymptotic
solution in \(K^3\) (resp. \(S\)).
Let \(\mathcal{O}_{K}^\times\) denote the unit group of \(\mathcal{O}_K\). We now prove that if \(A,B,C \in \{u2^r: u\in\mathcal{O}_K^\times, r \in
\mathbb{Z}_{\geq 0}\}\), then the generalized Fermat equation 1 has no asymptotic solutions in \(K_{n,l}^3\), for all integers \(n \geq
1\).
Theorem 4. Let \(K\) be a totally real number field of odd degree in which \(2\) is inert and write \(\mathfrak{P}=2\mathcal{O}_K\).
Let \(l \geq 5\) be a prime with \(l \nmid [K:\mathbb{Q}]\) and \(\gcd(\frac{l-1}{2}, [K:\mathbb{Q}])=1\). Let \(A,B,C \in \{u2^r:
u\in\mathcal{O}_K^\times, r \in \mathbb{Z}_{\geq 0}\}\). Assume
\(l\) is non-Wieferich, i.e., \(2^{l-1} \not\equiv 1 \pmod {l^2}\);
\(l\) is totally ramified in \(K\).
If \(A\pm B \pm C \neq 0\), \(\max\{v_\mathfrak{P}(A), v_\mathfrak{P}(BC)\} \leq 4\) and \(v_\mathfrak{P}(ABC) \equiv 0\) or \(2 \pmod 3\), then the generalized Fermat equation \(Ax^p+By^p+Cz^p=0\) has no asymptotic solution in \(K_{n,l}^3\), for all integers \(n \geq 1\). Moreover, if all elliptic curves \(E/K_{n,l}\) with full \(2\)-torsion are modular, then the generalized Fermat equation \(Ax^p+By^p+Cz^p=0\) has no effective asymptotic solution in \(K_{n,l}^3\).
Remark 5. Let \(A,B,C\),\(l\) and \(K\) be as in Theorem 4. Further, if \(A,B,C \in \mathbb{Z}\setminus \{0\}\), then Theorem 4 holds over each \(n\)-th layer \(K_{n,l}\) of the cyclotomic \(\mathbb{Z}_l\)-extension of \(K\) without the assumption \(A\pm B \pm C \neq 0\). This follows from [3].
For any number field \(F\), let \(h_F^+\) denote the narrow class number of \(F\). Let \(d\) be an odd prime. We now
study the solution of the generalized Fermat equation \(Ax^p+By^p+Cz^p=0\) over \(K\) in the case \(A,B,C \in \{\pm 2^rd^s: r,s \in \mathbb{Z}_{\geq
0}\}\).
Theorem 6. Let \(K\) be a totally real number field with \(2 \nmid h_K^+\) in which \(2\) is inert and write \(\mathfrak{P}=2\mathcal{O}_K\). Let \(d \geq 3\) be a prime with \(d \equiv 1 \pmod 4\). Assume
If \(A,B,C \in \{u 2^rd^s: u\in\mathcal{O}_K^\times, r,s \in \mathbb{Z}_{\geq 0}\}\), then the generalized Fermat equation \(Ax^p+By^p+Cz^p=0\) has no asymptotic solution \((a,b,c) \in \mathcal{O}_K^3\) with \(\mathfrak{P}|abc\). Moreover, if all elliptic curves \(E/K\) with full \(2\)-torsion are
modular, then the equation \(Ax^p+By^p+Cz^p=0\) has no effective asymptotic solution \((a,b,c) \in \mathcal{O}_K^3\) with \(\mathfrak{P}|abc\).
Remark 7. In [3], the author proved the generalized Fermat equation \(Ax^p+By^p+Cz^p=0\) has no asymptotic
solution \((a,b,c) \in \mathcal{O}_K^3\) with \(\mathfrak{P}|abc\) whenever \(A,B,C \in \{u 2^r: u\in\mathcal{O}_K^\times, r\in \mathbb{Z}_{\geq 0}\}\).
However, in Theorem 6, we use class field theory to extend the results of [3] to the case \(A,B,C \in \{u 2^rd^s: u\in\mathcal{O}_K^\times, r,s \in \mathbb{Z}_{\geq 0}\}\), where \(d\) is an odd prime.
Let \(d\) be an odd prime. We conclude this section by stating the following result. More precisely, we prove that the generalized Fermat equation 1 has no effective asymptotic
solutions over the \(n\)-th layer \(\mathbb{Q}_{n,l}\) of the cyclotomic \(\mathbb{Z}_l\) extension of \(\mathbb{Q}\)
whenever \(A,B,C \in \{\pm 2^rd^s: r,s \in \mathbb{Z}_{\geq 0}\}\).
Theorem 8. Let \(d\geq 3\) and \(l \geq 5\) be distinct rational primes, and let \(n \geq 1\) be a positive integer. Assume
\(l\) is non-Wieferich, i.e., \(2^{l-1} \not\equiv 1 \pmod {l^2}\);
If \(A,B,C \in \{\pm 2^rd^s: r,s \in \mathbb{Z}_{\geq 0}\}\) and \(2 \nmid h_{\mathbb{Q}_{n,l}}^+\), then the generalized Fermat equation \(Ax^p+By^p+Cz^p=0\) has no effective asymptotic solution \((a,b,c) \in \mathcal{O}_{\mathbb{Q}_{n,l}}^3\) with \(2|abc\), i.e., there exists an effective
computable constant \(V:=V_{K,A,B,C}\) (depending on \(K,A,B,C\)) such that for primes \(p >V\), the equation \(Ax^p+By^p+Cz^p=0\) has no non-trivial primitive solutions \((a,b,c) \in \mathcal{O}_{\mathbb{Q}_{n,l}}^3\) with \(2 |abc\).
Remark 9. The effectivity in Theorem 8 follows from a result of Thorne, which asserts that for every prime \(l\geq
2\) and every positive integer \(n \geq 1\), all elliptic curves over \(\mathbb{Q}_{n,l}\) are modular (cf. [7] for details).
2 The unit equation over cyclotomic \(\mathbb{Z}_l\) extensions of \(F\)↩︎
Let \(l\) be a rational prime and let \(n\geq 1\) be a positive integer. Let \(\mu_n\) be a primitive \(n\)-th root of
unity. Then the cyclotomic field \(\mathbb{Q}(\mu_{l^{n+1}})\) is a cyclic extension of \(\mathbb{Q}\) with degree \([\mathbb{Q}(\mu_{l^{n+1}}):
\mathbb{Q}]=l^n(l-1)\). Let \(\mathbb{Q}_{n,l}\) denote the unique subfield of \(\mathbb{Q}(\mu_{l^{n+1}})\) of degree \(l^n\) over \(\mathbb{Q}\). Let \(\mathbb{Q}_{\infty,l}:= \bigcup_{i=1}^{\infty} \mathbb{Q}_{n,l}\). Then \(\mathbb{Q}_{\infty,l}\) is the cyclotomic \(\mathbb{Z}_l\) extension of \(\mathbb{Q}\), and we call \(\mathbb{Q}_{n,l}\) as the \(n\)-th layer of \(\mathbb{Q}_{\infty,l}\). Let \(F\) be any number field. We write \(F_{\infty,l}:=F \cdot \mathbb{Q}_{\infty,l}\) the compositum of \(F\) and \(\mathbb{Q}_{\infty,l}\). Then \(F_{\infty,l}\) is a cyclotomic \(\mathbb{Z}_l\) extension of \(F\), and we call \(F_{n,l}:=F \cdot \mathbb{Q}_{n,l}\) the \(n\)-th layer of \(F_{\infty,l}\).
The following theorem is very useful in the proof of the main results. More precisely, we prove that the unit equation over \(F_{n,l}\) has no solutions for all \(n \geq 1\).
Theorem 10. Let \(F\) be a number field of degree \(m\), and let \(l\) be a prime that is totally ramified in \(F\). If either \(l=2\) or \(l\geq 5\) with \(l \nmid m\) and \(\gcd(\frac{l-1}{2}, {m})=1\),
then the unit equation \[\lambda + \mu=1, \;\lambda, \mu \in \mathcal{O}_{F_{n,l}}^\times\] has no solutions for all integers \(n \geq 1\).
Proof. Suppose the unit equation \(\lambda + \mu=1\) has a solution \(\;(\lambda, \mu) \in \mathcal{O}_{F_{n,l}}^\times \times \mathcal{O}_{F_{n,l}}^\times\) for some \(n \in \mathbb{N}\). Since \(l\) is totally ramified in both \(F\) and \(\mathbb{Q}_{n,l}\), it follows that \(l\) is totally ramified in \(M=F_{n,l}\). Let \(\mathfrak{q}\) denote the unique prime of \(M\) lying above \(l\). This implies \(l\mathcal{O}_M=\mathfrak{q}^{[M: \mathbb{Q}]}\) and \(\mathcal{O}_M/\mathfrak{q}\simeq \mathbb{F}_l\). Hence, there exists \(a \in \mathbb{Z}\) such that \(\lambda \equiv a \pmod \mathfrak{q}\). Now, we show that \(a^{[M: \mathbb{Q}]} \equiv \pm 1 \pmod l\).
Let \(L\) denote the normal closure of \(M/\mathbb{Q}\). Then \(l\mathcal{O}_L=(\mathfrak{q}\mathcal{O}_L)^{[M: \mathbb{Q}]}\). For every \(\sigma \in \mathrm{Gal}(L/ \mathbb{Q})\), we have \[(\sigma(\mathfrak{q}\mathcal{O}_L))^{[M: \mathbb{Q}]}=\sigma(l\mathcal{O}_L)=l\mathcal{O}_L=(\mathfrak{q}\mathcal{O}_L)^{[M: \mathbb{Q}]}.\]
Using the unique factorization of ideals in the Dedekind domain \(\mathcal{O}_L\), we conclude that \(\sigma(\mathfrak{q}\mathcal{O}_L)=\mathfrak{q}\mathcal{O}_L\) for all \(\sigma \in \mathrm{Gal}(L/ \mathbb{Q})\). This gives \(\sigma(\lambda) \equiv a \pmod {\mathfrak{q}\mathcal{O}_L}\) for all \(\sigma \in \mathrm{Gal}(L/
\mathbb{Q})\). Let \(\lambda_1, \dots, \lambda_{[M: \mathbb{Q}]}\) be the roots of the characteristic polynomial \(C_{M,\lambda}(x)\) in \(L\). Then
\(\lambda_i\)’s are the conjugates of \(\lambda\), i.e., \(\lambda_i= \sigma (\lambda)\) for some \(\sigma \in \mathrm{Gal}(L/
\mathbb{Q})\). This gives \(\lambda_i \equiv a \pmod {\mathfrak{q}\mathcal{O}_L}\), therefore we have \[C_{M,\lambda}(x)= (x-\lambda)\dots (x-\lambda_{[M: \mathbb{Q}]}) \equiv (x-a)^{[M:
\mathbb{Q}]} \pmod {\mathfrak{q}\mathcal{O}_L[x]}.\] Since \(C_{M,\lambda}(x), \;(x-a)^{[M: \mathbb{Q}]} \in \mathbb{Z}[x]\), we get \(C_{M,\lambda}(x) \equiv (x-a)^{[M: \mathbb{Q}]} \pmod
{l\mathbb{Z}[x]}\). This gives \(N_{M/\mathbb{Q}}(\lambda) \equiv a^{[M: \mathbb{Q}]} \pmod l\). Hence \(a^{[M: \mathbb{Q}]} \equiv \pm 1 \pmod l\).
If \(l=2\), then \(a^{[M: \mathbb{Q}]} \equiv 1 \pmod 2\) and hence \(a \equiv 1 \pmod 2\). This gives \(\lambda \equiv 1 \pmod
\mathfrak{q}\). Similarly \(\mu \equiv 1 \pmod \mathfrak{q}\). Hence \[1= \lambda+ \mu \equiv 1+1\pmod \mathfrak{q}.\] This is not possible since \(\mathfrak{q}|2\).
We now assume \(l \geq 5\). Then \(a^{\frac{l-1}{2}} \equiv \pm 1 \pmod l\). Recall that \([F:\mathbb{Q}]=m\) and \(l \nmid
m\). This implies \([M: \mathbb{Q}]=l^nm\). Using the assumption \(\gcd(m, \frac{l-1}{2})=1\), we get \(\gcd( [M: \mathbb{Q}], \frac{l-1}{2})= \gcd(l^nm,
\frac{l-1}{2})=1\). Hence there exists integers \(b,c\) such that \(b[M: \mathbb{Q}]+ c\frac{l-1}{2}=1\). This gives \[a =a^{b[M: \mathbb{Q}]}.a^{c
\frac{l-1}{2}} \equiv \pm 1 \pmod l.\] Hence \[\label{congruence32of32norm} \lambda \equiv \pm 1 \pmod \mathfrak{q}.\tag{2}\] Similarly, we get \(\mu \equiv \pm 1 \pmod \mathfrak{q}\). This gives \[1= \lambda+ \mu \equiv \pm1 \pm 1 \pmod \mathfrak{q}.\] This gives \(1 \equiv \pm1 \pm 1 \pmod l\), which is
not possible since \(l \geq 5\). ◻
3 Generalized Fermat equation over cyclotomic \(\mathbb{Z}_l\) extensions of \(K\)↩︎
In this section, we will prove the main results. Let \(K\) be a totally real number field, \(P_K\) denote the set of all non-zero prime ideals of \(\mathcal{O}_K\) and \(S_K:= \{ \mathfrak{P}\in P_K :\;\mathfrak{P}|2\}\). For any \(A,B,C \in \mathcal{O}_K\setminus \{0\}\), let \(S_K^{\prime}:= \{ \mathfrak{P}\in P_K :\;\mathfrak{P}|2ABC \}\). For any set \(S \subseteq P_K\), let \(\mathcal{O}_{S}:=\{\alpha \in K : v_\mathfrak{P}(\alpha)\geq 0
\text{ for all } \mathfrak{P}\in P \setminus S\}\) denote the ring of \(S\)-integers of \(K\) and \(\mathcal{O}_{S}^\times\) denote the group of \(S\)-units of \(K\).
The following criterion of the asymptotic generalized Fermat equation is a special case of [3], and will play an important role in the proofs of Theorems 1 and 4.
Theorem 11. Let \(K\) be a totally real number field of odd degree in which \(2\) is inert and write \(\mathfrak{P}=2\mathcal{O}_K\).
Let \(A,B,C \in \mathcal{O}_K\setminus \{0\}\). Suppose every solution \((\lambda, \mu)\) to the \(S_K^\prime\)-unit equation \[\label{S95K-unit32solution} \lambda+\mu=1, \;\lambda, \mu \in \mathcal{O}_{S_K^\prime}^\times\qquad{(1)}\] satisfies \[\label{assumption32for32main32result32x94p43y94p61294rz94p} \max \left\{|v_\mathfrak{P}(\lambda)|,|v_\mathfrak{P}(\mu)| \right\}\leq 4, \text{ and }
v_\mathfrak{P}(\lambda\mu)\equiv 1\pmod 3.\qquad{(2)}\] If \(A\pm B \pm C \neq 0\), \(\max\{v_\mathfrak{P}(A), v_\mathfrak{P}(BC)\} \leq 4\) and \(v_\mathfrak{P}(ABC) \equiv 0\) or \(2 \pmod 3\), then the generalized Fermat equation \(Ax^p+By^p+Cz^p=0\) has no asymptotic solution in \(K^3\). Moreover, if all elliptic curves \(E/K\) with full \(2\)-torsion are modular, then the equation \(Ax^p+By^p+Cz^p=0\) has
no effective asymptotic solution in \(K^3\).
The following lemma is a key ingredient in the proof of Theorems 1 and 4.
Lemma 12. Let \(K\) be a totally real number field of odd degree. Let \(l \geq 5\) be a prime that is totally ramified in \(K\) with
\(l \nmid [K:\mathbb{Q}]\) and \(\gcd(\frac{l-1}{2}, [K:\mathbb{Q}])=1\). Assume \(2\) is inert in \(F= K_{n,l}\). Write
\(\mathfrak{P}=2\mathcal{O}_F\) and \(S_F=\{\mathfrak{P}\}\). Then for every solution \((\lambda, \mu)\) to the \(S_F\)-unit
equation \(\lambda+\mu=1, \;\lambda, \mu \in \mathcal{O}_{S_F}^\times\) satisfies \(\left(v_\mathfrak{P}(\lambda), v_\mathfrak{P}(\mu)\right) \in \{(1,0), (0,1), (-1,-1)\}\).
Proof. We first show that \(v_\mathfrak{P}(\lambda) <2\). If not, let \(v_\mathfrak{P}(\lambda) \geq 2\). This gives \(\lambda \in
\mathcal{O}_F\) and \(\lambda \equiv 0 \pmod {\mathfrak{P}^2}\). Since \(\lambda+\mu=1\), we get \(v_\mathfrak{P}(\mu)=0\) and \(\mu \equiv 1 \pmod {\mathfrak{P}^2}\). This gives \(\mu \in \mathcal{O}_F^\times\) and \(\text{N}_{F/\mathbb{Q}}(\mu) \equiv 1 \pmod 4\), hence \(\text{N}_{F/\mathbb{Q}}(\mu)=1\). Since \(l\) is totally ramified in both \(K, \mathbb{Q}_{n,l}\), it follows that \(l\) is
totally ramified in \(F=K_{n,l}\). Let \(\mathfrak{q}\) be the unique prime of \(F\) lying above \(l\). From 2 , it follows that \(\mu \equiv \pm 1 \pmod \mathfrak{q}\). If \(\mu \equiv 1 \pmod \mathfrak{q}\), then \(\lambda \equiv 0
\pmod \mathfrak{q}\), which is not possible since \(\lambda \in \mathcal{O}_{S_F}^\times\) and \(\mathfrak{q}\notin S_F\). Hence \(\mu \equiv -1 \pmod
\mathfrak{q}\). Similar to the proof of Theorem 10, we get \(1=\text{N}_{F/\mathbb{Q}}(\mu)
\equiv (-1)^{[F: \mathbb{Q}] }\pmod l\). Since \([K : \mathbb{Q}]\) is odd and \(l\) is odd, it follows that \([F: \mathbb{Q}]\) is odd. Thus, we have
\(1 \equiv -1\pmod l\), which is not possible since \(l \geq 5\).
Next, we show that \(v_\mathfrak{P}(\lambda) >-2\). If not, let \(v_\mathfrak{P}(\lambda) \leq -2\). This gives \(v_\mathfrak{P}(\lambda)=v_\mathfrak{P}(\mu)\). Choose \(\lambda'=\frac{1}{\lambda}\) and \(\mu'= \frac{-\mu}{\lambda}\). Therefore, \(\lambda', \mu' \in \mathcal{O}_{S_F}^\times\), \(\lambda'+\mu'=1\) and \(v_\mathfrak{P}(\lambda') \geq 2\), which contradicts the previous
case. Hence, we have \(-2 < v_\mathfrak{P}(\lambda) <2\) and by symmetry \(-2 <v_\mathfrak{P}(\mu) <2\). Finally, if \(v_\mathfrak{P}(\lambda)
=0=v_\mathfrak{P}(\mu)\), then \(\lambda, \mu \in \mathcal{O}_F^\times\), which is not possible by Theorem 10. This finishes the proof of the lemma. ◻
We now recall a result of [6] that provides a necessary and sufficient condition for a prime to be inert in \(\mathbb{Q}_{n,l}\).
Lemma 13. [6] Let \(l \geq 3\) and \(d\geq 2\) be distinct rational
primes. Then \(d\) is inert in \(\mathbb{Q}_{n,l}\) if and only if \(d^{l-1} \not\equiv 1 \pmod {l^2}\).
Proof of Theorem 1. Recall that \(K_{n,l}=K \cdot \mathbb{Q}_{n,l}\) and \([\mathbb{Q}_{n,l} :
\mathbb{Q}]=l^n\). Since \(l \nmid [K:\mathbb{Q}]\), we get \(\gcd([K: \mathbb{Q}], [\mathbb{Q}_{n,l} : \mathbb{Q}])=1\) and \([K_{n,l} : \mathbb{Q}]=l^n[K:
\mathbb{Q}]\). Since \(2^{l-1} \not\equiv 1 \pmod {l^2}\), by Lemma 13, it
follows that \(2\) is inert in \(\mathbb{Q}_{n,l}\). Since \(2\) is inert in \(K\) and \(\gcd([K:
\mathbb{Q}], [\mathbb{Q}_{n,l} : \mathbb{Q}])=1\), we conclude that \(2\) is inert in \(F=K_{n,l}\). By Lemma 12, it follows that every solution \((\lambda, \mu)\) to the \(S_{F}\)-unit equation \(\lambda+\mu=1,
\;\lambda, \mu \in \mathcal{O}_{S_F}^\times\) satisfies \(\left(v_\mathfrak{P}(\lambda), v_\mathfrak{P}(\mu)\right) \in \{(1,0), (0,1), (-1,-1)\}\). Since \(K\) and \(\mathbb{Q}_{n,l}\) are totally real, it follows that \(K_{n,l}\) is totally real. Since both \([K: \mathbb{Q}]\) and \(l\) are
odd, we get \([K_{n,l}:\mathbb{Q}]\) is odd for all integers \(n \geq 1\). Finally, applying Theorem 11 with \(K=K_{n,l}\) and \(A=B=C=1\), the proof of the theorem follows. ◻
Proof of Theorem 4. Using the same arguments in the proof of Theorem 1,
we conclude that \(2\) is inert in \(F=K_{n,l}\) and the \(S_F\)-unit equation \(\lambda+\mu=1, \;\lambda, \mu \in
\mathcal{O}_{S_F}^\times\) satisfies \(\left(v_\mathfrak{P}(\lambda), v_\mathfrak{P}(\mu)\right) \in \{(1,0), (0,1), (-1,-1)\}\). Since \(A,B,C \in \{u2^r: u\in\mathcal{O}_K^\times, r \in
\mathbb{Z}_{\geq 0}\}\), we get \(S_{K_{n,l}}^\prime=S_{K_{n,l}}\) for all integers \(n \geq 1\). Since \(K_{n,l}\) is totally real and \([K_{n,l}: \mathbb{Q}]\) is odd for all integers \(n \geq 1\), the proof of the theorem follows from Theorem 11. ◻
The following criterion of the asymptotic generalized Fermat equation is a special case of [3], and will play an important role in the proof of Theorems 6 and 8.
Theorem 14. Let \(K\) be a totally real number field in which \(2\) is inert and write \(\mathfrak{P}=2\mathcal{O}_K\). Let \(A,B,C \in \mathcal{O}_K\setminus \{0\}\). Suppose every solution \((\lambda, \mu)\) to the \(S_K^\prime\)-unit equation \(\lambda+\mu=1, \;\lambda, \mu \in \mathcal{O}_{S_K^\prime}^\times\) satisfies \[\label{assumption32on32S32unit32crit32even} \max
\left\{|v_\mathfrak{P}(\lambda)|,|v_\mathfrak{P}(\mu)| \right\}\leq 4.\qquad{(3)}\] Then, the generalized Fermat equation \(Ax^p+By^p+Cz^p=0\) has no asymptotic solution \((a,b,c) \in
\mathcal{O}_K^3\) with \(\mathfrak{P}|abc\). Moreover, if all elliptic curves \(E/K\) with full \(2\)-torsion are modular, then the equation \(Ax^p+By^p+Cz^p=0\) has no effective asymptotic solution \((a,b,c) \in \mathcal{O}_K^3\) with \(\mathfrak{P}|abc\).
The following proposition is a key ingredient in the proof of Theorems 6 and 8. More precisely, we prove that for any number field \(K\) with \(S_K^{\prime}= \{ \mathfrak{P}\in P_K :\;\mathfrak{P}|2d\}\), every solution \((\lambda, \mu)\) to the \(S_K^\prime\)-unit equation \(\lambda+\mu=1, \;\lambda, \mu \in \mathcal{O}_{S_K^\prime}^\times\) satisfies ?? .
Proposition 15. Let \(K\) be a number field in which \(2\) is inert and
write \(\mathfrak{P}=2\mathcal{O}_K\). Let \(d\geq 3\) be a prime and \(S_K^{\prime}= \{ \mathfrak{P}\in P_K :\;\mathfrak{P}|2d\}\). Assume
\(2 \nmid h_K^+\);
\(d \equiv 1 \pmod 4\) and \(d\) is inert in \(K\);
the congruence \(d \equiv v^2 \mod \mathfrak{P}^5\) with \(v\in \mathcal{O}_K\) has no solutions.
Then every solution \((\lambda, \mu)\) to the \(S_K^\prime\)-unit equation \(\lambda+\mu=1, \;\lambda, \mu \in \mathcal{O}_{S_K^\prime}^\times\)
satisfies \[\max \left\{|v_\mathfrak{P}(\lambda)|,|v_\mathfrak{P}(\mu)| \right\}\leq 4.\]
To prove Proposition 15, we need the following lemma, which is a special case of [8].
Lemma 16. Let \(K\) be a number field and \(S_K^{\prime}= \{ \mathfrak{P}\in P_K :\;\mathfrak{P}|2d\}\). Let \((\lambda, \mu)\) be a
solution to the \(S_K^\prime\)-unit equation \(\lambda+\mu=1, \;\lambda, \mu \in \mathcal{O}_{S_K^\prime}^\times\) and let \(\mathfrak{P}\in S_{K}\). Then
there exists \(\lambda', \mu' \in \mathcal{O}_{S_K^\prime}^\times\) with \(v_\mathfrak{P}(\lambda') \geq 0\) and \(v_\mathfrak{P}(\mu') \geq
0\) such that \(\lambda'+\mu'=1\) and \(\max \left\{|v_\mathfrak{P}(\lambda)|,|v_\mathfrak{P}(\mu)| \right\}= \max \left\{v_\mathfrak{P}(\lambda'),v_\mathfrak{P}(\mu')
\right\}\).
Proof of Proposition 15. The proof of this proposition follows arguments similar to
those used in the proofs of [9] and [10]. We need to show that \(\max \left\{|v_\mathfrak{P}(\lambda)|,|v_\mathfrak{P}(\mu)| \right\}\leq 4\) for every solution \((\lambda, \mu)\) to the \(S_K^\prime\)-unit equation \(\lambda+\mu=1, \;\lambda, \mu \in \mathcal{O}_{S_K^\prime}^\times\). Suppose there exists a solution \((\lambda, \mu)\) with \(\max
\left\{|v_\mathfrak{P}(\lambda)|,|v_\mathfrak{P}(\mu)| \right\}\geq 5\). Without loss of generality by Lemma 16, we can take \(v_\mathfrak{P}(\lambda) \geq 0\) and \(v_\mathfrak{P}(\mu) \geq 0\), hence \(\max \left\{v_\mathfrak{P}(\lambda),v_\mathfrak{P}(\mu)\right\}\geq 5\). Without
loss of generality, take \(v_\mathfrak{P}(\lambda) \geq 5\). Since the \(S_K^\prime\)-unit equation \(\lambda+\mu=1\) admits only finitely many solutions, we
choose \((\lambda^\prime, \mu^\prime=1-\lambda^\prime)\) such that \(v_\mathfrak{P}(\lambda^\prime)\) is maximal among all such solutions. Therefore \(v_\mathfrak{P}(\lambda') \geq 5\), hence \(v_\mathfrak{P}(\mu') =0\) and \(\mu'\equiv 1\pmod {\mathfrak{P}^5}\).
We now show that \(\mu'\) is a square in \(\mathcal{O}_{S_{K}'}^\times\). Since \(v_\mathfrak{P}(\mu')=0\), we get
\[\label{factor32of32mu} \mu'= \alpha d^s,\tag{3}\] for some \(\alpha \in \mathcal{O}_{K}^\times\) and \(s\in
\mathbb{Z}\). First, we show that \(\alpha\) is a square. Suppose \(\alpha\) is not a square in \(\mathcal{O}_{K}\). Consider the field \(L=K(\sqrt{\alpha})\). Then \([L:K]=2\). Since \(d \equiv 1 \pmod {4}\), we have \(d ^s \equiv 1 \pmod {4}\). From 3 , we deduce that \(\alpha \equiv 1 \pmod {\mathfrak{P}^2}.\) and hence \(\frac{1-\alpha}{4} \in \mathcal{O}_K\). Then \(L=K(\sqrt{\alpha})=K(\beta)\), where \(\beta=\frac{1-\sqrt{\alpha}}{2}\). Then the minimal polynomial \(m_\beta(x)= x^2-x+ \frac{1- \alpha}{4} \in
\mathcal{O}_{K}[x]\) with its discriminant \(\alpha \in \mathcal{O}_{K}^\times\). Therefore, \(L\) is unramified at all the finite places of \(K\) and
\([L:K]=2\), which contradicts the hypothesis \(2\nmid h_{K}^+\). Hence, \(\alpha\) must be a square.
Next, we show that \(s\) is even. Suppose \(s\) is odd. Let \(s=2k+1\) for some \(k \in \mathbb{Z}\). From 3 , we get \(\mu '= (\alpha d^{2k})d\). Since \(\alpha\) is a square and \(\mu'\equiv 1\pmod {\mathfrak{P}^5}\), it
follows that \(d \equiv v^2\pmod {\mathfrak{P}^5}\), for some \(v \in \mathcal{O}_K\), which contradicts the hypothesis \((3)\). Therefore,, \(s\) is even. Hence, \(\mu'\) is a square in \(\mathcal{O}_{S_{K}'}^\times\).
Let \(\mu'=\gamma^2\) for some \(\gamma \in \mathcal{O}_{S_{K}'}^\times\). Then \(v_\mathfrak{P}(\mu') =v_\mathfrak{P}(\gamma)=0\). So \[\lambda'= 1-\mu'=1-\gamma^2= (1+\gamma)(1-\gamma).\] Choose \(s_0= v_\mathfrak{P}(\lambda')\), \(s_1= v_\mathfrak{P}(1+\gamma)\) and \(s_2= v_\mathfrak{P}(1-\gamma)\). This gives \(s_0= s_1+s_2\geq 5\), \(s_1 \geq 0\) and \(s_2 \geq 0\). By a simple calculation,
we get either \(s_1=1\) or \(s_2=1\). Without loss of generality, we take \(s_1=1\). This gives \(s_2= s_0-1 \geq 4\).
Choose \(\lambda''= \frac{-(1-\gamma)^2}{4\gamma}\) and \(\mu''= \frac{(1+\gamma)^2}{4\gamma}\). Then \(\lambda'' + \mu
''=1\). So \((\lambda'', \mu'')\) is a solution to the \(S_{K}'\)-unit equation \(\lambda+\mu=1\). We have \(v_\mathfrak{P}(\lambda'')=2 s_2-2=2 s_0-4>s_0= v_\mathfrak{P}(\lambda')\), which is not possible since \(v_\mathfrak{P}(\lambda')\) is maximal among all the solutions to the
\(S_{K}'\)-unit equation. This finishes the proof of the proposition. ◻
The following lemma is very useful in the proof of Theorems 6 and 8.
Lemma 17. Let \(F\) be a number field and \(n \in \mathbb{N}\). Assume \(2\) is inert in \(F\) and
write \(\mathfrak{P}=2\mathcal{O}_F\). For any \(\lambda, \nu \in \mathcal{O}_{F}\) , if \(\lambda \equiv \nu \pmod {\mathfrak{P}^n}\), then \(\text{N}_{F/\mathbb{Q}}(\lambda) \equiv \text{N}_{F/ \mathbb{Q}}(\nu) \pmod {2^n}\).
Proof. Let \(M\) be the normal closure of \(F/\mathbb{Q}\). Then for every \(\sigma \in \mathrm{Gal}(M/ \mathbb{Q})\), we have \(\sigma(\lambda )\equiv \sigma(\nu) \pmod {\mathfrak{P}^n}\). This gives \(\text{N}_{F/\mathbb{Q}}(\lambda) \equiv \text{N}_{F/ \mathbb{Q}}(\nu) \pmod {\mathfrak{P}^n}\). Since \(\text{N}_{F/\mathbb{Q}}(\lambda), \text{N}_{F/ \mathbb{Q}}(\nu) \in \mathbb{Z}\), the proof of the lemma follows. ◻
Proof of Theorem 6. Since \(A,B,C \in \{\pm 2^rd^s: r,s \in \mathbb{Z}_{\geq 0}\}\), it follows that \(S_K^{\prime}= \{ \mathfrak{P}\in P_K :\;\mathfrak{P}|2d\}\). To prove this theorem, it suffices to verify that all the hypotheses of Proposition 15 are satisfied. Clearly, the hypothesis \((1)\), \((2)\) are satisfied. Suppose the
hypothesis \((3)\) is not true, i.e., \(d \equiv v^2 \mod \mathfrak{P}^5\) for some \(v\in \mathcal{O}_K\). By Lemma 17, we get \(d^{[K: \mathbb{Q}]} \equiv a^2 \pmod {32}\), for some \(a \in \mathbb{Z}\). Since the only odd squares modulo
\(32\) are \(\{1,9, 17, 25\}\), we get \(d^{[K: \mathbb{Q}]} \equiv 1\) or \(9\) or \(17\)
or \(25\)\(\pmod {32}\), which contradicts the hypothesis \((3)\) of Theorem 6. Thus by Proposition 15, it follows that every solution
\((\lambda, \mu)\) to the \(S_K^\prime\)-unit equation \(\lambda+\mu=1, \;\lambda, \mu \in \mathcal{O}_{S_K^\prime}^\times\) satisfies \[\max \left\{|v_\mathfrak{P}(\lambda)|,|v_\mathfrak{P}(\mu)| \right\}\leq 4.\] Finally, the proof of the theorem follows from Theorem 14. ◻
Proof of Theorem 8. Since, \(A,B,C \in \{\pm 2^rd^s: r,s \in \mathbb{Z}_{\geq 0}\}\), we get \(S_K^{\prime}= \{ \mathfrak{P}\in P_K :\;\mathfrak{P}|2d\}\). To prove this theorem, it suffices to verify that all the hypotheses of Proposition 15 are satisfied for \(K= \mathbb{Q}_{n,l}\). Since \(2^{l-1}\not\equiv 1 \pmod {l^2}\) and
\(d^{l-1} \not\equiv 1 \pmod {l^2}\), it follows from Lemma 13 that both \(2\) and \(d\) are inert in \(\mathbb{Q}_{n,l}\). Hence the hypotheses \((1)\), \((2)\) of
Proposition 15 are satisfied for \(K=\mathbb{Q}_{n,l}\).
Suppose the hypothesis \((3)\) is not true \(K=\mathbb{Q}_{n,l}\), i.e., \(d \equiv v^2 \mod \mathfrak{P}^5\) for some \(v\in
\mathcal{O}_{\mathbb{Q}_{n,l}}\), where \(\mathfrak{P}=2\mathcal{O}_{\mathbb{Q}_{n,l}}\). By Lemma 17, we get \(d^{l^n} \equiv a^2 \pmod {32}\), for some \(a \in \mathbb{Z}\). Since the only odd squares modulo \(32\) are \(\{1,9, 17,
25\}\), we get \(d^{l^n} \equiv 1\) or \(9\) or \(17\) or \(25\)\(\pmod {32}\).
This gives \(d^{l^n} \equiv 1 \pmod 8\). Since \(l^n\) is odd, we have \(d \equiv d^{l^n} \equiv 1 \pmod 8\). This implies \(d
\equiv 1\) or \(9\) or \(17\) or \(25\)\(\pmod {32}\), contradicting the hypothesis \((4)\) of Theorem 8.
Therefore, all the hypotheses of Proposition 15 are satisfied for \(K=\mathbb{Q}_{n,l}\). Thus, every solution \((\lambda, \mu)\) to the \(S_K^\prime\)-unit equation \(\lambda+\mu=1, \;\lambda, \mu \in
\mathcal{O}_{S_K^\prime}^\times\) satisfies \[\max \left\{|v_\mathfrak{P}(\lambda)|,|v_\mathfrak{P}(\mu)| \right\}\leq 4.\] Finally, the proof of Theorem 8 follows by applying Theorem 14 to the case \(K=\mathbb{Q}_{n,l}\) and \(A,B,C \in \{\pm 2^rd^s: r,s \in \mathbb{Z}_{\geq 0}\}\). ◻