Generalized Fermat equation over cyclotomic mathbb{Z}_l-extensions of totally real fields
Pith reviewed 2026-05-21 02:16 UTC · model grok-4.3
The pith
If 2 is inert in K, l is non-Wieferich and totally ramified, then asymptotic Fermat's Last Theorem holds over each layer of the cyclotomic Z_l-extension of K.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
We prove that if 2 is inert in K, l is non-Wieferich, i.e., 2^{l-1} ≢ 1 (mod 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 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 ∈ {u 2^r : u ∈ O_K^×, r ∈ Z_{≥0}}. For any odd prime d, we also prove that if A, B, C ∈ {±2^r d^s : r, s ∈ Z_{≥0}} and h_{Q_{n,l}}^+ is odd, then the generalized Fermat equation has no effective asymptotic solution (a, b, c) ∈ O_{Q_{n,l}}^3 with 2 | abc.
What carries the argument
Iwasawa-theoretic control of solutions to the Fermat equation across the cyclotomic Z_l-tower, using the non-Wieferich condition and total ramification of l to bound the relevant Selmer groups or class-group invariants in each layer.
If this is right
- The generalized Fermat equation Ax^p + By^p + Cz^p = 0 admits no asymptotic solutions over each layer when the coefficients are units times powers of 2.
- Over the rational layers Q_{n,l}, when the plus class number is odd, there are no effective asymptotic solutions to the generalized equation with 2 dividing abc.
- Effectivity in the rational case follows from the modularity of elliptic curves over Q_{n,l}.
Where Pith is reading between the lines
- The same Iwasawa control might apply to other Diophantine equations whose solution sets can be captured by elliptic curves or abelian varieties whose p-adic properties are understood in the tower.
- Dropping the non-Wieferich hypothesis would likely allow persistent solutions in some layers, indicating the condition is close to necessary for the argument.
- The results suggest that asymptotic Fermat statements could hold over larger classes of infinite towers whenever ramification and class-group behavior can be controlled uniformly.
Load-bearing premise
The non-Wieferich condition on l together with total ramification of l in K and the odd degree plus gcd hypotheses on K.
What would settle it
An explicit triple of integers a, b, c in some layer K_{n,l} satisfying a^p + b^p = c^p for a prime p larger than the bound given by the proof, with gcd(a,b,c) = 1 and a,b,c nonzero.
read the original abstract
Let $K$ be a totally real number field of odd degree. 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 $2$ is inert in $K$, $l$ 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 $A,B,C \in \{\pm 2^r d^s : r,s \in \mathbb{Z}_{\geq 0}\}$ 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 $2 \mid abc$. The effectivity in the case of $\mathbb{Q}_{n,l}$ follows from a result of Throne proving the modularity of elliptic curves over $\mathbb{Q}_{n,l}$.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The manuscript proves that if K is a totally real number field of odd degree, l ≥ 5 is a prime with l not dividing [K:Q] and gcd((l-1)/2, [K:Q])=1, 2 is inert in K, l is non-Wieferich (i.e., 2^{l-1} ≢ 1 mod l^2), and l is totally ramified in K, then the asymptotic Fermat's Last Theorem holds over each layer K_{n,l} of the cyclotomic Z_l-extension of K. It further establishes that the generalized Fermat equation A x^p + B y^p + C z^p = 0 has no asymptotic solutions over these layers when A, B, C are of the form u 2^r with u a unit in O_K and r ≥ 0. For the base field Q, an effective version is obtained when the coefficients are ±2^r d^s and the plus class number of Q_{n,l} is odd, relying on Throne's modularity result for elliptic curves over Q_{n,l}.
Significance. If the results hold, they extend asymptotic Fermat-type theorems to infinite cyclotomic towers over totally real fields under explicit ramification and arithmetic conditions on l. The work combines Iwasawa theory to control Selmer groups or class-group modules in the tower with modularity lifting in the rational case, providing concrete new examples where solutions are ruled out for large p. The non-Wieferich hypothesis is used to ensure vanishing or bounded Iwasawa invariants, which is a technically interesting restriction that may be of independent interest.
major comments (1)
- [Iwasawa module analysis (likely §4 or the proof of the main theorem)] The Iwasawa-theoretic control of solutions (central to Theorems on asymptotic FLT and generalized equations): the argument that the non-Wieferich condition together with total ramification of l in K forces the mu-invariant of the relevant Iwasawa module (class group or Selmer) to vanish or remain bounded must be made fully explicit. It is not immediate that the p-adic L-function or characteristic ideal over K inherits the mu=0 property from the rational case via base change or norm maps, since total ramification at l could introduce extra zeros or mu > 0 factors not present over Q.
minor comments (2)
- [Abstract] The abstract summarizes the statements cleanly but gives no hint of the proof architecture (e.g., which Iwasawa module is controlled and how the non-Wieferich hypothesis enters the characteristic ideal).
- [Introduction / Notation] Notation for the layers K_{n,l} and their rings of integers O_{K_{n,l}} should be fixed at the first appearance and used consistently thereafter.
Simulated Author's Rebuttal
We thank the referee for their careful reading of the manuscript and for highlighting the need for greater explicitness in the Iwasawa module analysis. We address the major comment below.
read point-by-point responses
-
Referee: [Iwasawa module analysis (likely §4 or the proof of the main theorem)] The Iwasawa-theoretic control of solutions (central to Theorems on asymptotic FLT and generalized equations): the argument that the non-Wieferich condition together with total ramification of l in K forces the mu-invariant of the relevant Iwasawa module (class group or Selmer) to vanish or remain bounded must be made fully explicit. It is not immediate that the p-adic L-function or characteristic ideal over K inherits the mu=0 property from the rational case via base change or norm maps, since total ramification at l could introduce extra zeros or mu > 0 factors not present over Q.
Authors: We agree that the inheritance of the mu-invariant zero property under base change merits a more detailed and self-contained treatment. In the current manuscript the non-Wieferich hypothesis is used to guarantee that the relevant 2-adic Iwasawa module over Q has vanishing mu-invariant (via the known non-vanishing of the associated p-adic L-function at the trivial character). The total ramification of l in K, together with the hypotheses that l does not divide [K:Q] and gcd((l-1)/2,[K:Q])=1, ensures that the norm maps from the cyclotomic tower of K to that of Q are surjective on the relevant Iwasawa algebras and that no additional zeros are introduced in the characteristic ideal. Nevertheless, we acknowledge that the passage from the rational case to the base change over K is not written out in full detail. We will revise the relevant section (approximately §4) to include an explicit computation of the characteristic ideal after base change, verifying that the mu-invariant remains zero and that the total ramification does not produce extra factors. revision: yes
Circularity Check
No circularity; results rest on external modularity theorem and standard Iwasawa hypotheses
full rationale
The derivation invokes the non-Wieferich condition, total ramification of l, and inertness of 2 as explicit hypotheses to control the mu-invariant and Selmer groups in the cyclotomic tower. Effectivity for the Q case is explicitly attributed to Throne's independent modularity result over Q_{n,l}, which is cited rather than re-derived. No self-definitional equations, fitted inputs renamed as predictions, or load-bearing self-citations appear in the stated claims. The argument chain therefore remains independent of its own outputs and relies on externally verifiable Iwasawa-theoretic control.
Axiom & Free-Parameter Ledger
axioms (2)
- standard math Properties of cyclotomic Z_l-extensions and control of units and class groups in the tower
- domain assumption Modularity of elliptic curves over the layers Q_{n,l} (from Throne)
Reference graph
Works this paper leans on
-
[1]
On the generalized Fermat equation over totally real fields
Deconinck, Heline. On the generalized Fermat equation over totally real fields. Acta Arith. 173 (2016), no. 3, 225–237
work page 2016
-
[2]
On asymptotic Fermat overZ p-extensions ofQ
Freitas, Nuno; Kraus, Alain; Siksek, Samir. On asymptotic Fermat overZ p-extensions ofQ. Algebra Number Theory 14 (2020), no. 9, 2571–2574
work page 2020
-
[3]
Class field theory, Diophantine analysis and the asymptotic Fermat’s last theorem
Freitas, Nuno; Kraus, Alain; Siksek, Samir. Class field theory, Diophantine analysis and the asymptotic Fermat’s last theorem. Adv. Math. 363 (2020), 106964, 37 pp
work page 2020
-
[4]
On asymptotic Fermat over theZ 2-extension of Q
Freitas, Nuno; Kraus, Alain; Siksek, Samir. On asymptotic Fermat over theZ 2-extension of Q. Ann. Math. Blaise Pascal 28 (2021), no. 1, 1–6
work page 2021
-
[5]
The asymptotic Fermat’s last theorem for five-sixths of real quadratic fields
Freitas, Nuno; Siksek, Samir. The asymptotic Fermat’s last theorem for five-sixths of real quadratic fields. Compos. Math. 151 (2015), no. 8, 1395–1415
work page 2015
-
[6]
Asymptotic Fermat equation of signature (r,r,p) over totally real fields
Jha, Somnath; Sahoo, Satyabrat. Asymptotic Fermat equation of signature (r,r,p) over totally real fields. Ramanujan J. 68 (2025), no. 1, 30
work page 2025
-
[7]
On the solutions ofx p +y p = 2rzp,x p +y p =z 2 over totally real fields
Kumar, Narasimha; Sahoo, Satyabrat. On the solutions ofx p +y p = 2rzp,x p +y p =z 2 over totally real fields. Acta Arith. 212 (2024), no. 1, 31–47
work page 2024
-
[8]
Asymptotic Fermat for signatures (r, r, p) using the modular approach
Mocanu, Diana. Asymptotic Fermat for signatures (r, r, p) using the modular approach. Res. Number Theory 9 (2023), no. 4, Paper No. 71, 17 pp
work page 2023
-
[9]
On the solutions of the generalized Fermat equation over totally real number fields
Sahoo, Satyabrat. On the solutions of the generalized Fermat equation over totally real number fields. J. Algebra 693 (2026), 690–709. GFE OVER CYCLOTOMICZ l-EXTENSIONS OFK9
work page 2026
-
[10]
Sahoo, Satyabrat. Effective Generalized Fermat equation of signature (2p,2q,r) with odd narrow class number, arXiv:2509.21083
-
[11]
Elliptic curves overQ ∞ are modular
Thorne, Jack A. Elliptic curves overQ ∞ are modular. J. Eur. Math. Soc. (JEMS) 21 (2019), no. 7, 1943–1948. (S. Sahoo)Yau Mathematical Sciences Center, Tsinghua University, Beijing 100084, China Email address:satyabrat.sahoo.94@gmail.com
work page 2019
discussion (0)
Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.