On Asymptotic Fermat Over $\Mathbb {Z} P $ Extensions of $\Mathbb {Q} $

$On Asymptotic Fermat Over $\Mathbb {Z} P $ Extensions of $\Mathbb {Q} $$

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Last Theorem. In this paper we use elementary arguments to study these S-unit equations in Qn,p and this study, together with the S-unit criterion, quickly yields Theorem 2. The eﬀectivity in Theorem 2 builds on the following great theorem due to Thorne [5].

Theorem (Thorne). Elliptic curves over Qn,p are modular.

2. An S-unit criterion for asymptotic FLT The following criterion for asymptotic FLT is a special case of [2, Theorem 3]. Theorem 3. Let F be a totally real number ﬁeld. Suppose the Eichler–Shimura conjecture over F holds. Assume that 2 is inert in F and write q = 2OF for the × prime ideal above 2. Let S = {q} and write OS for the group of S-units in F . Suppose every solution (λ, µ) to the S-unit equation × (2.1) λ + µ = 1, λ,µ ∈OS satisﬁes both of the following conditions

(2.2) max{Sordq(λ)S, Sordq(µ)S} ≤ 4, ordq(λµ) ≡ 1 (mod 3). Then the asymptotic Fermat’s Last Theorem holds over F . Moreover, if all elliptic curves over F with full 2-torsion are modular, then the eﬀective asymptotic Fermat’s Last Theorem holds over F . For a discussion of the Eichler–Shimura conjecture see [2, Section 2.4], but for the purpose of this paper we note that the conjecture is known to hold for all totally real ﬁelds of odd degree. In particular, it holds for Qn,p for all odd p. To apply Theorem 3 to F = Qn,p we need to know for which p is 2 inert in F . The answer is given by the following lemma, which for n = 1 is Exercise 2.4 in [6].

Lemma 2.1. Let p ≥ 3, q be distinct primes. Then q is inert in Qn,p if and only if qp−1 ~≡ 1 (mod p2).

Proof. Let L = Q(ζpn+1 ) and F = Qn,p. Write σq and τq for the Frobenius elements corresponding to q in Gal(L~Q) and Gal(F ~Q). The prime q is inert in F precisely n when τq has order p . The natural surjection Gal(L~Q) → Gal(F ~Q) sends σq to τq and its kernel has order p − 1. Thus q is inert in F if and only if the order of σq n p−1 n is divisible by p , which is equivalent to σq having order p . There is a canonical n+1 × n+1 isomorphism Gal(L~Q) → (Z~p Z) sending σq to q + p Z. Thus q is inert in F if and only if qp−1 +pn+1Z has order pn. This is equivalent to qp−1 ~≡ 1 (mod p2).

3. Proof of Theorem 2

Lemma 3.1. Let p be the unique prime above p in F = Qn,p. Let λ ∈ OF . Then λ ≡ NormF ~Q(λ) (mod p).

Proof. As p is totally ramiﬁed in F , we know that the residue ﬁeld OF ~p is Fp. Thus there is some a ∈ Z such that λ ≡ a (mod p). Let σ ∈ G = Gal(F ~Q). Since pσ = p, we have λσ ≡ a (mod p). Hence σ #G NormF ~Q(λ) = M λ ≡ a (mod p). σ∈G n However #G = p so NormF ~Q(λ) ≡ a ≡ λ (mod p). ASYMPTOTIC FERMAT 3

Lemma 3.2. Let p ≠ 3 be a rational prime. Let F = Qn,p. Then the unit equation × (3.1) λ + µ = 1, λ,µ ∈OF has no solutions. Proof. Let (λ, µ) be a solution to (3.1). By Lemma 3.1, λ ≡ ±1 (mod p) and µ ≡ ±1 (mod p). Thus ±1 ± 1 ≡ λ + µ = 1 (mod p). This is impossible as p ≠ 3. Remark 3.3. Lemma 3.2 is false for p = 3. Indeed, Let p = 3 and n = 1. Then 3 2 F = Q1,3 = Q(θ) where θ satisﬁes θ −6θ +9θ−3 = 0. The unit equation has solution λ = 2 − θ and µ = −1 + θ. In fact, the unit equation solver of the computer algebra system Magma [1] gives a total of 18 solutions.

Lemma 3.4. Let p ≥ 5 be a rational prime. Let F = Qn,p. Suppose 2 is inert in F × and write q = 2OF for the unique prime above 2. Let S ={q} and write OS for the group of S-units. Then every solution to the S-unit equation (2.1) satisﬁes one of the following:

(i) ordq(λ) = 1, ordq(µ) = 0; (ii) ordq(λ) = 0, ordq(µ) = 1; (iii) ordq(λ) = ordq(µ) = −1.

Proof. Write nλ = ordq(λ) and nµ = ordq(µ). Suppose first nλ ≥ 2. Then nµ = 0 × σ and so µ ∈OF . Moreover, as 4 S λ, we have µ ≡ 1 (mod 4) and so µ ≡ 1 (mod 4) for σ all σ ∈ G = Gal(F ~Q). Hence NormF ~Q(µ) = ∏ µ ≡ 1 (mod 4). But NormF ~Q(µ) = ±1, thus NormF ~Q(µ) = 1. As before, denote the unique prime above p by p. By Lemma 3.1 we have µ ≡ 1 (mod p). Hence p divides 1−µ = λ giving a contradiction. ′ ′ Thus nλ ≤ 1. Next suppose nλ ≤ −2. Then nλ = nµ. Let λ = 1~λ and µ = ′ ′ −µ~λ. Then (λ ,µ ) is a solution to the S-unit equation satisfying nλ′ ≥ 2, giving a contradiction by the previous case. Hence −1 ≤ nλ ≤ 1 and by symmetry −1 ≤ nµ ≤ 1. From Lemma 3.2 either nλ ≠ 0 or nµ ≠ 0. Thus one of (i), (ii), (iii) must hold. Remark 3.5. Possibilities (i), (ii), (iii) cannot be eliminated because of the solutions (2, −1), (−1, 2) and (1~2, 1~2) to the S-unit equation. Proof of Theorem 2. We suppose p ≥ 5 and non-Wieferich. It follows from Lemma 2.1 that 2 is inert in F = Qn,p. Write q = 2OF . By Lemma 3.4 all solutions (λ, µ) to the S-unit equation (2.1) satisfy (2.2). We now apply Theorem 3. As elliptic curves over Qn,p are modular thanks to Thorne’s theorem, we conclude that the effective Fermat’s Last Theorem holds over Qn,p. Remark 3.6. The proof of Theorem 2 for p = 3 and for the Wieferich primes seems out of reach at present. There are solutions to the unit equation in Q1,3 (as indicated in Remark 3.3), and therefore in Qn,3 for all n, and these solutions violate the criterion of Theorem 3. For p a Wieferich prime, 2 splits in Qn,p into at least p prime ideals and we would need to consider the S-unit equation (2.1) with S the set of primes above 2. It appears difficult to treat the S-unit equation in infinite families of number fields where #S ≥ 2 (c.f. [3, Theorem 7] and its proof).

4. A Generalization In fact, the proof of Theorem 2 establishes the following more general theorem. Theorem 4. Let F be a totally real number ﬁeld and p ≥ 5 be a rational prime. Suppose that the following conditions are satisﬁed. 4 NUNOFREITAS,ALAINKRAUS,ANDSAMIRSIKSEK

(a) F is a p-extension of Q (i.e. F ~Q is a Galois extension of degree pn for some n ≥ 1). (b) p is totally ramiﬁed in F . (c) 2 is inert in F . Then the asymptotic Fermat’s Last Theorem holds for F . Example 4.1. A quick search on the L-Functions and Modular Forms Database [4] yields 153 ﬁelds of degree 5 satisfying conditions of the theorem with p = 5. The one with smallest discriminant is Q1,5. The one with the next smallest discriminant is F = Q(θ) where θ5 − 110θ3 − 605θ2 − 990θ − 451 = 0. The discriminant of F is 8 4 5 ⋅ 11 . It is therefore not contained in any Zp-extension of Q. References

[1] W. Bosma, J. Cannon et C. Playoust: The Magma Algebra System I: The User Language, J. Symb. Comp. 24 (1997), 235–265. (see also http://magma.maths.usyd.edu.au/magma/) 3.3 [2] N. Freitas and S. Siksek, The asymptotic Fermat’s Last Theorem for five-sixths of real qua- dratic fields, Compositio Mathematica 151 (2015), 1395–1415. 1, 2, 2 [3] N. Freitas, A. Kraus and S. Siksek, Class field theory, Diophantine analysis and the asymptotic Fermat’s Last Theorem, Advances in Mathematics 363 (2020), to appear. 1, 3.6 [4] The LMFDB Collaboration, The L-functions and Modular Forms Database, http://www.lmfdb.org, 2020, [Online; accessed 24 January 2020]. 4.1 [5] J. Thorne, Elliptic curves over Q∞ are modular, Journal of the European Mathematical Society 21 (2019), 1943–1948. 1 [6] L. C. Washington, Introduction to Cyclotomic Fields, second edition, GTM 83, Springer, 1997. 2 [7] A. Wieferich, Zum letzten Fermatschen Theorem, Journal für die reine und angewandte Math- ematik 136 (1909), 293–302. 1 [8] A. Wiles, Modular elliptic curves and Fermat’s Last Theorem, Annals of Math. 141 (1995), 443–551. 1

Departament de Matemàtiques i Informàtica, Universitat de Barcelona (UB), Gran Via de les Corts Catalanes 585, 08007 Barcelona, Spain E-mail address: [email protected]

Sorbonne Université, Institut de Mathématiques de Jussieu - Paris Rive Gauche, UMR 7586 CNRS - Paris Diderot, 4 Place Jussieu, 75005 Paris, France E-mail address: [email protected]

Mathematics Institute, University of Warwick, CV4 7AL, United Kingdom E-mail address: [email protected]