Nicolas Billerey

Imin Chen

Lassina Dembélé

Luis Dieulefait

Nuno Freitas

SOME EXTENSIONS OF THE MODULAR METHOD AND FERMAT EQUATIONS OF SIGNATURE (13, 13, n)

Keywords: . 2010 Mathematics Subject Classification. Primary 11D41; Secondary 11G10, 11F80 Fermat equations, abelian surfaces, modularity, Galois representations

We provide several extensions of the modular method which were motivated by the problem of completing previous work to prove that, for any integer n ≥ 2, the equation

has no non-trivial primitive solutions. In particular, we present four elimination techniques which are based on: (1) establishing reducibility of certain residual Galois representations over a totally real field; (2) generalizing image of inertia arguments to the setting of abelian surfaces; (3) establishing congruences of Hilbert modular forms without the use of often impractical Sturm bounds; and (4) a unit sieve argument which combines information from classical descent and the modular method.

The extensions are of broader applicability and provide further evidence that it is possible to obtain a complete resolution of a family of generalized Fermat equations by remaining within the framework of the modular method. As a further illustration of this, we complete a theorem of Anni-Siksek to show that, for , m ≥ 5, the only primitive solutions to the equation x 2 + y 2m = z 13 are trivial.

Introduction

In [START_REF] Billerey | A multi-Frey approach to Fermat equations of signature (r, r, p)[END_REF], it was shown that it is possible to get an optimal result for a family of generalized Fermat equations, specifically x 5 + y 5 = 3z n for n ≥ 2 an integer, using a refined modular method with the multi-Frey approach over totally real fields. These methods were also applied to the equation (1.1)

x 13 + y 13 = 3z p , where p is prime, but failed for p = 7; this failure is due to the presence of five specific Hilbert modular forms after level lowering which the authors were not able to eliminate. Four of the obstructing forms are newforms of weight 2 defined over the real cubic subfield of Q(ζ 13 ), the cyclotomic field of 13th root of unity, whose level is the unique ideal of norm 2808 = 2 3 ⋅3 3 ⋅13. The remaining one, which is the most relevant to us, is denoted by g; it is a newform defined over Q( √ 13) of parallel weight 2, level N = (2 3 13) and field of coefficients Q( √ 2) appearing in [4, Proposition 9].

The techniques in this paper provide extensions of the modular method which were initially motivated by the problem of eliminating these five obstructing forms, but are of broader applicability. In particular, we successfully treat the remaining case p = 7 of equation (1.1). Our first step is to reduce the resolution of (1.1) for p = 7 to the problem of dealing with the form g. For this, we consider the Frey curve F = F a,b defined in [4, §7.2]. The refined elimination technique from [4, §7.3] applied to F shows that the obstruction to the modular method arises from the four obstructing newforms defined over the real cubic subfield of Q(ζ 13 ), with level of norm 2808. We are then able to show that each of these forms has a reducible mod p representation for some prime p 7 (see Proposition 3.2). Since the mod 7 representation attached to F is irreducible by assumption, this allows us to discard these four forms (see Sections 2 and 4). However, establishing the reducibility property for the four obstructing forms requires a bit of care since the usual approach, which involves using the Sturm bound, is not computationally feasible in the setting of Hilbert modular forms.

The remaining obstructing form g arises for the Frey curve E = E a,b defined in [4, §7.1], more specifically, when trying to prove [START_REF] Billerey | A multi-Frey approach to Fermat equations of signature (r, r, p)[END_REF]Theorem 7] for p = 7. We will introduce three different methods for discarding this form.

The first method eliminates g by extending an 'image of inertia argument' to the setting of abelian surfaces. For this, we will combine modularity of an abelian surface with the study of the inertial types of g at suitable primes.

The second method is to show that g is congruent modulo 7 to the Hilbert newform associated to the Frey curve corresponding to the trivial solution (1, -1, 0); see also [START_REF] Billerey | A multi-Frey approach to Fermat equations of signature (r, r, p)[END_REF]Remark 7.4]. In general, the proof of such congruences requires computing Hecke eigenvalues up to the Sturm bound, which is often not practical for Hilbert modular forms, as is the case in our situation. Instead, we exploit a multiplicity one phenomenon for the residual Galois representations attached to the Hilbert newforms in our situation, to establish the desired congruence.

The third method is based on combining the study of units in classical descent with local information coming from the modular method together with restrictions on the solutions coming from the multi-Frey approach in [START_REF] Billerey | A multi-Frey approach to Fermat equations of signature (r, r, p)[END_REF].

As a consequence, we obtain a complete resolution of the generalized Fermat equations in (1.1). More precisely, we establish the following result.

Theorem 1.1. For all integers n ≥ 2, there are no integer solutions (a, b, c) to the equation x 13 + y 13 = 3z n such that abc ≠ 0 and gcd(a, b, c) = 1.

An integer solution (a, b, c) to (1.1) is said to be a primitive solution if gcd(a, b, c) = 1, and trivial if abc = 0.

It is natural to wonder whether one could use Chabauty methods to solve (1.1) for p = 7 by determining the rational points on the hyperelliptic curves associated with (1.1) from the constructions in [START_REF] Dahmen | Perfect powers expressible as sums of two fifth or seventh powers[END_REF]. This would necessitate computing the 2-Selmer groups of Jacobians of genus 3 hyperelliptic curves of the form y 2 = x 7 +a, where a belongs to the maximal totally real sextic subfield F of the field Q(ζ 13 ). Because some values of a which need to be treated are not seventh powers in F , such calculations would require working over the extensions F ( 7 √ a)

which are of absolute degree 42. Even under GRH, this would be extremely challenging with current methods. Moreover, it would only lead to a conditional resolution of (1.1).

The computations required to support the proof of our theorems were performed using Magma [START_REF] Bosma | The Magma algebra system. I. The user language[END_REF] (V2.25-5), mainly the Hilbert Modular Forms Package (see [16,[START_REF] Bushnell | The local Langlands conjecture for GL(2)[END_REF]). The program and output transcript files are available at [3] (see Read_Me.txt for a description of the files). The Dokchister-Doris package for computing the conductor of genus 2 curves is available at [START_REF] Doris | Package for computing the conductor or genus 2 curves[END_REF].

Acknowledgements. We would like to thank T. Dokchitser and C. Doris for making their algorithm available to us, and for helpful email correspondence. We also thank M. Stoll for conversations regarding the use of the Chabauty method for the problem treated in this paper, A. Pacetti for discussions concerning Sturm bounds for Hilbert modular forms and N. Mascot for (double-)checking part (1) of Theorem 5.3 using the algorithm in [START_REF] Costa | Rigorous computation of the endomorphism ring of a Jacobian[END_REF]. We finally thank the anonymous referees for their careful reading and useful remarks.

The elimination step away from reducible primes

In this section, we recall the elimination step of the modular method and discuss ways in which it can fail to yield a contradiction. The discussion will center on equation (1.1) for convenience, but the same principles apply more generally.

Let (a, b, c) be a primitive solution to (1.1) and E a,b K a Frey curve attached to it, where K is a totally real number field. Also, let ρ E a,b ,p be the mod p representation attached to E a,b .

By assumption ρ E a,b ,p is an irreducible representation which is modular. An application of standard results on level lowering implies that there is a Hilbert newform f of level N and weight 2, with field of coefficients Q f , such that

(2.1) ρ E a,b ,p ≃ ρ f,p
where p p is a prime in Q f and ρ f,p the mod p representation attached to f . In practice, the level N is a concrete ideal dividing the conductor of E a,b , independent of (a, b, c) and of 'small' norm. So the new subspace S 2 (N) new of Hilbert cusp forms of level N, weight 2 and trivial character is accessible via the Hilbert Modular Forms package in Magma [START_REF] Bosma | The Magma algebra system. I. The user language[END_REF].

Given f ∈ S 2 (N) new , a candidate newform, a crucial step in the modular method consists of showing that the isomorphism (2.1) does not occur for any prime p of Q f above p. We say that we have eliminated the form f for the exponent p if we can successfully complete that step. By eliminating all the newforms in S 2 (N) new , we obtain a contradiction to the fact that ρ E a,b ,p is modular, thus solving (1.1) for the exponent p.

By taking traces at Frobenius elements on both sides of (2.1) one gets

(2.2) a q (E a,b ) ≡ a q (f ) (mod p) for all primes q ∤ p ⋅ cond(E a,b ),
where a q (f ) is the Hecke eigenvalue of f at q, and a q (E a,b ) the trace of Frobenius of E at q.

Let q ≠ p be a rational prime coprime to the level N of f . Suppose that E a,b has good reduction at each prime ideal q dividing q in K, and define the quantity

B q (E a,b , f ) = gcd ({Norm (a q (E a,b ) -a q (f )) ∶ q q}) .
Note that (2.2) implies the exponent p divides qB q (E a,b , f ).

Suppose instead that E a,b has multiplicative reduction at a prime q q. Then, the isomorphism (2.1) implies that the form f satisfies level raising conditions, hence we have the congruence:

(2.3) a q (f ) ≡ ±(Norm(q) + 1) (mod p).

In the elimination step, we find rational primes q such that, for all a, b mod q not both zero, the Frey curve E a,b has either good reduction at all q q or multiplicative reduction at all q q. Then, for a newform f ∈ S 2 (N) new , we define

A q (f ) ∶= q (0,0)≠(a,b)∈F 2 q B q (E a,b , f ) ⋅ q q Norm(a q (f ) 2 -(Norm(q) + 1) 2 ),
where the first part accounts for the possibility of E having good reduction at q whilst the second product for the case of multiplicative reduction. The quantity A q (f ) is independent of p, so once we find one prime q such that A q (f ) ≠ 0, we have eliminated f for large p. For most f , trying a few auxiliary primes often suffices to find such a q. This is then enough to eliminate f for all exponents p ∤ A q (f ). Furthermore, by using several primes q 1 , . . . , q s , we eliminate f for all p ∤ gcd(A q 1 (f ), . . . , A qs (f )).

For a given newform f , the elimination step leads to a set of primes P f , which cannot be excluded by this process. This set usually consists of a few small primes, but it can also be empty or equal to the set of the prime numbers; the latter only occurs when all the auxiliary primes we tried gave A q (f ) = 0.

In [4, §7.3], four of the authors introduced the following refined elimination technique to deal with the remaining primes in P f . The key observation of this refinement is that (2.2) and (2.3) imply p A q (f ) but the converse is not true. So, for the primes p ∈ P f , they test for the congruences (2.2) and (2.3), which result from the isomorphism (2.1), directly.

In other words, for each prime p p in Q f , they check the following congruences by direct computations:

(i) for all q q of good reduction for E a,b we have a q (f ) ≡ a q (E a,b ) (mod p);

(ii) for all q q of multiplicative reduction for E a,b , we have a q (f ) ≡ ±(Norm(q)+1) (mod p);

This can be done as long as we can factor the primes p ∈ P f in the field Q f . If (i) and (ii) fail for p p then we have eliminated the pair (f, p). Excluding all such pairs then eliminates f for the exponent p. In [4, §7.3], this refinement allowed for the resolution of (1.1) for p = 5, 11.

Let us now discuss one way in which the argument above can fail. Suppose for some form f and a prime p 0 p ∈ P f in Q f , the mod p 0 representation is reducible and, moreover, it satisfies ρ ss f,p 0 ≃ χ p ⊕ 1, where χ p is the mod p cyclotomic character. Then, this form satisfies a q (f ) ≡ Norm(q) + 1 (mod p 0 ), and therefore congruence (ii) holds with p = p 0 . Thus we are unable to eliminate the pair (f, p 0 ) using the refined elimination technique.

However, we note that the isomorphism (2.1) was obtained via level lowering theorems, which require the representations involved to be residually irreducible. So, in testing for the congruences (i) and (ii), we only need to consider those pairs (f, p), where p p is a prime in Q f such that ρ f,p is irreducible. This allows us to circumvent the reducibility issue raised in the previous paragraph. We will see in Section 3 that one of the reasons why equation (1.1) does not follow directly from the methods in [START_REF] Billerey | A multi-Frey approach to Fermat equations of signature (r, r, p)[END_REF], for p = 7, is the presence of reducible pairs (f, p 0 ). We note that this idea of avoiding reducible primes p 0 appeared in the work of Bugeaud-Mignotte-Siksek [7], where they work with classical forms over Q. In that setting, a practical Sturm bound is available, allowing them to prove isomorphism of the form ρ ss f,p 0 ≃ χ p ⊕ 1. For our purpose, due to the lack of practical Sturm bounds for Hilbert modular forms, we will establish the existence of our reducible pairs (f, p 0 ) using base change arguments.

A reducibility result for Hilbert modular forms

In this section, we let K = Q(b) be the real cubic subfield of Q(ζ 13 ), where b 3 + b 2 -4b + 1 = 0. We let q 13 be the unique ramified prime in K, which lies above 13. We note that 2 and 3 are inert in K.

Let N = 2 ⋅ 3 ⋅ q 13 , and write S 2 (N) new for the new subspace of Hilbert cusp forms of level N, weight 2 and trivial character. For f ∈ S 2 (N) new we let Q f denote the field of coefficients of f . For a newform f and p p a prime in Q f , write ρ f,p ∶ G K → GL 2 (F p ) the mod p representation attached to f . Recall that ρ f,p can ramify only at primes dividing N and p.

For certain newforms f ∈ S 2 (N) new , we will establish the reducibility of ρ f,p , where p 7 is a ramified prime in Q f . We remark that, for cubic fields (or higher degree fields), there are no effective Sturm bounds available in the literature. Furthermore, even in the quadratic case, such effective bounds are too large to be of any practical use (see Section 6).

Instead, we present a method for establishing reducibility in certain situations when the residual representation is a base change from Q. The method is presented in the particular case of interest, but could apply in other similar settings.

The space S 2 (N) new has dimension 181, and decomposes into 15 Hecke constituents as:

(3.1) 181 = 1 + 1 + 1 + 1 + 3 + 6 + 6 + 12 + 15 + 18 + 18 + 21 + 24 + 27 + 27.

More specifically, there are:

1. Four newforms with rational Hecke eigenvalues. One of them corresponds to the base change of the elliptic curve with Cremona label 78a1. The other three correspond to non base change elliptic curves over K which are permuted by the action of Gal(K Q). 2. One newform with Hecke eigenvalues in the real cubic field

Q(ζ 7 ) + ⊂ Q(ζ 7 ).
3. Ten newforms whose Hecke eigenvalue fields have degrees 6, 12, 15, 18, 21, 24 or 27. For each of the fields Q f , there is a unique subfield E such that Q f is a ray class field of degree 3 over E. We have summarized the structure of these Hecke eigenvalue fields in Table 1.

When [Q f ∶ Q] ≤ 21, we have given a generator for the conductor f Q f of Q f E, and its factorization. For [Q f ∶ Q] > 21,
only the the factorization is provided.

Let σ be a generator for Gal(K Q). Let f be a newform in a Hecke constituent in S 2 (N) new , and d f the dimension of this constituent. By cyclic base change and Shimura [44], there exists a newform

f σ ∈ S 2 (N) new such that Q f σ = Q f and a q (f σ ) = a q σ (f )
, for all primes q of K, Table 1. The Hecke eigenvalue fields for the newforms in S 2 (N) new over the cubic subfield K of Q(ζ 13 ) + which are a cubic extension of a unique subfield.

[Q f ∶ Q] = 6 E Q(w) ∶= Q[x] (x 2 -x -7) f Q f (2w + 2) = p 2 p 7 [Q f ∶ Q] = 6 E Q(w) ∶= Q[x] (x 2 -x -1) f Q f (-8w + 6) = p 2 p 19 [Q f ∶ Q] = 12 E Q(w) ∶= Q[x] (x 4 + 2x 3 -9x 2 -12x -3) f Q f (-w 2 -3w + 3) = p 2 p 7 [Q f ∶ Q] = 15 E Q(w) ∶= Q[x] (x 5 -47x 3 -105x 2 + 70x + 1) f Q f (1 8(-3w 4 -w 3 + 138w 2 + 369w + 9)) = p 2 [Q f ∶ Q] = 18 E Q(w) ∶= Q[x] (x 6 -x 5 -58x 4 -79x 3 + 856x 2 + 2865x + 2495) f Q f (-8w 5 + 32w 4 + 363w 3 -425w 2 -5425w -7264) = p 2 p 31 [Q f ∶ Q] = 18 E Q(w) ∶= Q[x] (x 6 -28x 4 -2x 3 + 104x 2 + 104x + 28) f Q f (1 2(-w 4 + 2w 3 + 26w 2 -48w -40)) = p 2 p 7 [Q f ∶ Q] = 21 E Q(w) ∶= Q[x] (x 7 + x 6 -58x 5 + 172x 4 + 84x 3 -744x 2 + 512x + 144) f Q f (1 8(-17w 6 -77w 5 + 726w 4 -296w 3 -2904w 2 + 2144w + 592)) = p 2 p 7 p ′ 7 [Q f ∶ Q] = 24 E Q(w) ∶= Q[x] (x 8 + 3x 7 -53x 6 -84x 5 + 1018x 4 + 190x 3 -6992x 2 + 5440x + 5060) f Q f p 2 p 7 [Q f ∶ Q] = 27 E Q(w) ∶= Q[x] (x 9 + 2x 8 -96x 7 -99x 6 + 2894x 5 + 1462x 4 -32500x 3 + 2240x 2 + 119964x -102212) f Q f p 2 p 7 p 13 [Q f ∶ Q] = 27 E Q(w) ∶= Q[x] (x 9 -x 8 -90x 7 + 106x 6 + 2878x 5 -4048x 4 -38316x 3 + 61316x 2 + 172200x -284688) f Q f p 2 p 1153
so that Gal(K Q) acts on the Hecke constituents. Since [K ∶ Q] = 3 the orbits of this action are either of size 1 or 3. The Hecke constituents of dimension 1 split into two orbits under this action: there is one orbit of size 1 which corresponds to the base change of the elliptic curve with Cremona label 78a1; the other orbit of size 3, underlined in (3.1), corresponds to a set of 3 elliptic curves permuted by Gal(K Q).

Assume that d f ≥ 3. Then, since the number of Hecke constituents of dimension d f ≥ 3 is at most 2, we conclude these constituents must be fixed by the action of Gal(K Q). Thus, f and f σ belong to the same constituent. However, a quick inspection of the first few Hecke eigenvalues shows that f is not a base change. Therefore, there exists a non-trivial element

τ ∈ Gal(Q f E) such that a q (f σ ) = a q σ (f ) = τ (a q (f ))
, for all primes q of K.

Lemma 3.1. For d = 6, 12, 18, 21, 24 or 27, let

S d ⊂ S 2 (N) new be the unique Hecke constituent of dimension d containing a newform f = f d with 7 Norm(f Q f ), where f Q f is as in Table 1.
Then, for every prime p 0 gcd(f Q f , (7)), we have Norm(p 0 ) = 7, and p 0 totally ramifies in

Q f E.
Proof. This follows from a computation in Magma.

Proposition 3.2. For d = 12, 21, 24 or 27, let f ∈ S d and p 0 in Q f be as in Lemma 3.1. Suppose that ρ f,p 0 ramifies at all the primes dividing the level N. Then, ρ f,p 0 is reducible.

Proof. Let f and p 0 be as in the statement. Suppose that ρ f,p 0 is irreducible.

Recall from the discussion preceding Lemma 3.1 that for a non-trivial

σ ∈ Gal(K Q) there is τ ∈ Gal(Q f E) such that a q σ (f ) = τ (a q (f ))
, for all primes q.

Since p 0 is totally ramified in Q f E, and the residue field is

F p 0 = F 7 , τ ∶ Q f → Q f gives rise to the identity map F 7 → F 7 . It follows that (3.2) a q σ (f ) ≡ a q (f ) (mod p 0 ), for all primes q. Let σ ∈ Gal(Q Q) be a lift for σ. Consider the conjugate representation ρ σ f,p 0 which satisfies ρ σ f,p 0 (Frob q ) ∶= ρ f,p 0 (Frob q σ ) = ρ f,p 0 (σ Frob q σ-1 ).
(We note that these equalities are independent of the lift σ.) By Equation (3.2), we have

Tr(ρ σ f,p 0 (Frob q )) = a q σ (f ) ≡ a q (f ) ≡ Tr(ρ f,p 0 (Frob q )) (mod p 0 )
, for all primes q. Since ρ f,p 0 is irreducible (hence absolutely irreducible since p 0 ∤ 2), the representations ρ f,p 0 and ρ σ f,p 0 are isomorphic. From [25, Theorem 2.14, Chapter III], we conclude that there is an irreducible representation

ρ ∶ G Q → GL 2 (F 7 ) such that ρ G K = ρ f,p 0 .
We will now determine the Serre level N = N (ρ), the nebentypus (ρ) and Serre weight k(ρ).

We have k(ρ) = 2 because ρ G K = ρ f,p 0 is of parallel weight 2 and 7 is unramified in K.

Recall that 2 and 3 are inert in K, and that 13 is the only ramified prime in K. By assumption ρ f,p 0 ramifies at all primes q N = 2 ⋅ 3 ⋅ q 13 , hence ρ f,p 0 is a Steinberg representation at these primes (since q N). It follows that ρ is Steinberg at 2 and 3; thus υ (N (ρ)) = 1 for = 2, 3. At = 13, the representation ρ is either Steinberg or a twist of Steinberg by a character χ of G Q that becomes trivial over K. In the latter case, χ is of conductor 13 1 and the conductor of ρ at 13 is 13 2 . So twisting ρ by χ, if necessary, we can assume also that υ 13 (N (ρ)) = 1 since both ρ and ρ ⊗ χ restrict to ρ f,p 0 . Thus N (ρ) = 2 ⋅ 3 ⋅ 13 = 78.

Since ρ is Steinberg at = 2, 3, 13 it follows that the nebentypus (ρ) is locally trivial at ; thus (ρ) = 1 as there are no other ramified primes for ρ.

From the above discussion and Khare-Wintenberger (Serre's conjecture) [START_REF] Khare | On Serre's conjecture for 2-dimensional mod p representations of Gal(Q Q)[END_REF], ρ is modular ; hence, there is a classical newform h and a prime p 7 in Q h such that ρ ≃ ρ h,p . it follows that there is a newform h ∈ S 2 (78) new such that ρ ≃ ρ h,p for some prime p 7 in Q h .

There is only one such h, corresponding to the isogeny class of the elliptic curve W , with Cremona label 78a1, whose trace of Frobenius at q = 5 is a 5 (W ) = 2. The prime 5 splits in K so, for q 5, we must have the following congruences

Tr(ρ f,p 0 (Frob q )) = Tr(ρ(Frob q )) = Tr(ρ h,p (Frob q )) = Tr(ρ W,7 (Frob 5 )) ≡ 2 (mod 7).
But we easily check that we have ρ f,p 0 (Frob q ) ≡ 2 (mod 7).

This contradiction implies that ρ f,p 0 must be reducible, as desired.

Remark 3.3. Let f be a newform in the unique Hecke constituent S 18 such that 7 ramifies in Q f (as described in Table 1). Then, one easily checks that ρ f,p is irreducible for the unique prime p f Q f above 7. By adapting the proof of Proposition 3.2, one can show that f is in fact congruent to the base change of one of the newforms in S 2 (78, χ) new . In this case, we see that ρ f,p is unramified at q 13 . Remark 3.4. Let f ∈ S 2 (N) new be a newform as in Proposition 3.2. Then, f appears in the cohomology of a Shimura curve. For q = (2), (3) and q 13 , we have U q f = ±f , where U q is the Hecke operator at q. We believe that the reducibility of ρ f,p can be proved by generalising work of Martin [37] or of Ribet and Yoo [40,[START_REF] Yoo | Non-optimal levels of a reducible mod modular representation[END_REF] on non-optimal levels for reducible representations to the setting of Shimura curves over totally real number fields.

Reduction to the problem of eliminating the form g

We recall that in order to get a complete resolution of equation (1.1), it only remains to deal with the prime p = 7. There are five possible obstructing forms to this, four of which are described in Proposition 3.2 and Table 1; and the form g defined over Q( √ 13) described in the introduction. The goal of this section is to explain how we discard the four forms in Proposition 3.2 by using the refined elimination technique described in Section 2. We then deal with g in three different ways in Sections 5, 6, and 7.

Since we are following the same strategy as in [START_REF] Billerey | A multi-Frey approach to Fermat equations of signature (r, r, p)[END_REF], a crucial step in solving (1.1) for p = 7 is to extend [4, Theorem 7] to the exponent p = 7. To this end, we must prove the following: Let F = F a,b be the Frey curve defined over K, which was introduced in [4, §7.2]. Note that υ 2 (a + b) = υ 2 (3c p ) ≥ 3 and 3 a + b, therefore [4, Lemma 11] gives that ρ F,7 ≅ ρ f,p , for some f ∈ S 2 (N) new and p 7 in Q f . The elimination technique described in Section 2, applied with the quantities A q (f ) for the rational primes q = 5, 7, 11, 17, 31, shows that ρ F,7 ≅ ρ f,p except when f is as in Proposition 3.2. To complete the proof of the theorem, we make use of the refinement explained in Section 2 to rule out the remaining four forms. To this end, let f be as in Proposition 3.2 (see also Table 1), and suppose that ρ F,7 ≅ ρ f,p , where p 7 in Q f . Since ρ F,7 has conductor N, the same is true for ρ f,p . Furthermore, ρ f,p is irreducible (because ρ F,7 is irreducible by [4, Theorem 8]) so by Proposition 3.2 we can assume that p is unramified in Q f . Choosing a q = 2, 3, 13 satisfying q ≡ 1 (mod 13), we obtain from [4, Lemma 8] that (i) either q ∤ a + b and then for all q above q, we have a q (f ) ≡ a q (F a,b ) (mod p); (ii) or q a + b and then for all q above q, we have a q (f ) ≡ ±(Norm(q) + 1) (mod p).

By computing a q (F x,y ) for each q q and all x, y ∈ {0, . . . , q -1} not both zero, we eliminate f by checking that neither of the above congruences hold for all unramified primes p 7 in Q f . Indeed, the auxiliary prime q = 5 suffices to eliminate all the four possible f .

An image of inertia argument with an abelian surface

In this section, we give a proof of Theorem 4.1 based on generalizing an image of inertia argument to the setting of abelian surfaces.

From now on, we let K = Q(w) where w 2 = 13. We let

O K = Z[u], with u = 1+w
2 , be the ring of integers of K. We consider the hyperelliptic curve C defined over K by

C ∶ y 2 = (32u + 36)x 6 + (24u + 40)x 5 + (-u -32)x 4 + (-16u + 8)x 3 + (17u -28)x 2 (5.1) + (-6u + 16)x + 6u -16,
and we denote its Jacobian by J.

Lemma 5.1. The surface J has potentially good reduction at the primes (2) and (w), and we have cond(J) = N 2 , where N = (2 3 w 2 ).

Proof. We use Magma [START_REF] Bosma | The Magma algebra system. I. The user language[END_REF] to compute the odd part of the conductor of J, and the Dokchitser-Doris algorithm [START_REF] Dokchitser | Doris 3-torsion and conductor of genus 2 curves[END_REF] to get the even part. This yields that cond(J) = N 2 .

Let q be either of the primes (2) and (w) of K and consider C over K q . By [35, Théorème 1 (V)], there is a stable model C of C over an extension F of K q such that the special fiber of C is a union of two elliptic curves intersecting at a single point.

Let J be the Néron model of the Jacobian of C . By the discussion preceding [35, Proposition 2] and [35, Proposition 2(v)] itself, the special fibre of J is an abelian variety. Hence, J has potentially good reduction at the primes (2) and (w).

We record the following additional lemma for later, which also proves more concretely the assertion that J has potentially good reduction at (w).

Let K ′ = Q(ζ 13 +ζ -1
13 ) be the maximal totally real subfield of Q(ζ 13 ). Then K ′ K is a cyclic extension ramified at (w) only. Let K ′ w be the completion of K ′ at the unique prime above (w). Lemma 5.2. The surface J acquires good reduction over K ′ w .

Proof. Using Magma [START_REF] Bosma | The Magma algebra system. I. The user language[END_REF], the conductor exponent of J at the unique prime of K ′ above w is computed to be 0.

Let g be the Hilbert newform over K with parallel weight 2, trivial character and level N = (2 3 w 2 ) listed in [4, Proposition 9]. Since ord 2 (N) = 3 is odd, the local component π g,2 of the automorphic representation π g attached to g is supercuspidal. Therefore, the Eichler-Shimura conjecture for totally real fields holds (see [39, Proposition 2.20.2] or [49, Theorem B]). Thus, there is an abelian surface A g with RM by Z[ √ 2] attached to g. The theorem below shows that A g is isogenous to J. Theorem 5.3. Let C and J be given by (5.1). Then, we have the following:

(1) The ring End K (J) contains Z[ √ 2], i.e. J is of GL 2 -type with real multiplication (RM) by Q( √ 2); (2) The surface J is modular and corresponds to the Hecke constituent of the Hilbert newform g. In other words, J and A g are isogenous.

Proof. In [ Using Magma's implementation of Mestre's method [START_REF] Mestre | Construction de courbes de genre 2 a partir de leurs modules[END_REF], we can find a hyperelliptic curve C ′ of genus 2 defined over K whose Igusa-Clebsch invariants are the I ′ 2i above for i = 1, 2, 3, 5. Hence, A arises from the Jacobian of C ′ K and is thus itself defined over K.

Let I 2 , I 4 , I 6 and I 10 be the Igusa-Clebsch invariants of the curve C, and α = -60u-48. Then, we have I 2i = α 2i I ′ 2i for i = 1, 2, 3, 5. This shows that C K is isomorphic to C ′ K over K. The points in Y -(8) corresponding to (r ′ , s ′ ) are still rational over K, hence by the modular interpretation of Y -(8) there is a choice of A in its K-isomorphism class such that A is defined over K and A has RM by

Z[ √ 2]. Since Aut(C) ≅ Aut(C ′ ) is of order 2, we see C K and C ′ K differ by a quadratic twist. Thus, End K (J) also contains Z[ √ 2] by [45, Lemma 2.

2], hence proving (1).

Alternatively, the algorithm in [START_REF] Costa | Rigorous computation of the endomorphism ring of a Jacobian[END_REF] can be used to show the endomorphism ring End K (J) contains Z[ √ 2]. We thank N. Mascot and the authors of [START_REF] Costa | Rigorous computation of the endomorphism ring of a Jacobian[END_REF] for checking this.

We now prove [START_REF] Bennett | Ternary Diophantine equations via Galois representations and modular forms[END_REF]. Recall that 3 is inert in Q( √ 2), and consider the 2-dimensional 3-adic Galois representation attached to J

ρ J,3 ∶ Gal(Q K) → GL 2 (Q 3 ( √ 2)).
Since J is of GL 2 -type, it follows from Ribet [41, Proposition 3.4] (note that the proof generalizes to abelian varieties defined over totally real fields) that the determinant of ρ J,3 is the cyclotomic character, hence for q ∤ 3N its characteristic polynomial at q is of the form

charpoly(ρ J,3 (Frob q )) = x 2 -a q x + Norm(q),
where tr(ρ J,3 (Frob q )) = a q ∈ Q( √ 2), and Norm(q) is the norm of q.

Let ρJ,3 be the mod 3 reduction of this representation. By construction, ρJ,3 is odd and we see that the image of ρJ,3 is contained in

{u ∈ GL 2 (F 9 ) ∶ det(u) ∈ F × 3 } and, since -1 is a square in F ×
9 , the projective image lands in PSL 2 (F 9 ). By computing the orders of the conjugacy classes of ρJ,3 (Frob q ) for the primes q above 17 and 53, we see that the projective image of ρJ,3 contains elements of orders 2, 4 and 5. There is no proper subgroup of PSL 2 (F 9 ) which contains three elements with those orders, hence the projective image of ρJ,3 is PSL 2 (F 9 ). In particular, we see that the image of ρJ,3 contains SL 2 (F 9 ), so ρJ,3 is absolutely irreducible.

The prime 3 splits in K. Writing (3) = v 1 v 2 , where v 1 = (u -1) and v 2 = (u), we get that tr(ρ J,3 (Frob v 1 )) = 2 ± √ 2, and tr(ρ J,3 (Frob v 2 )) = ± √ 2 by computing the Euler factors of the curve C at both places and factoring over Q( √ 2). These traces are units modulo 3, so ρ J,3 is ordinary at each v 3. Further, since 5 ∤ N, we see that ρJ,3 I 5 is trivial, hence it has odd order. Since 3 and 5 have odd ramification indices in K, it follows that ρJ,3 satisfies the conditions of [24, Theorem 3.2 and Proposition 3.4]. Hence, it is modular.

We use [31, Theorem 3.5.5] to conclude that ρ J,3 , and hence J, is modular.

By local-global compatibility ([10, Théorème (A)]) and Lemma 5.1, the level of the Hilbert newform attached to J is N = (2 3 w 2 ). There is a unique Hecke constituent of weight 2 and level N whose Euler factors match those of the surface J, it is the one corresponding to the newform g.

The 'image of inertia argument' discards the possibility of the mod p representation of the Frey curve and that of a newform f being isomorphic by showing they have different image sizes at an inertia subgroup. This idea, originally from [33, p. 8], has been extensively applied and refined [START_REF] Bennett | Ternary Diophantine equations via Galois representations and modular forms[END_REF] (see [4, §3] for a description of two refinements) in the case of f corresponding to an elliptic curve. In its essence, this argument boils down to showing that the Frey curve and the newform f have different inertial types at some prime q dividing the level of f . Thus far, such inertia arguments have been restricted to the case of f corresponding to an elliptic curve because a method to explicitly determine the inertial types of a form f with non-rational coefficients has not been worked out in general.

In this section, we will use the modularity in Theorem 5.3 to describe (see Theorem 5.6) the inertial type of the non-rational form g at the prime (2); this together with the local information at (w) given by Lemma 5.2 allows for a proof of Theorem 1.1.

Before proceeding we need some notation. Let E = E 1,-1 be the Frey curve attached to the trivial solution (1, -1, 0) in [4, Section 7.1]; it admits a minimal model given by

E ∶ y 2 = x 3 -ux 2 + (9u -25)x -17u + 49.
Let also J be as above and p 7 be the prime of Q( √ 2) above 7 generated by 3 + √ 2.

Let K 2 Q 2 be the unique unramified quadratic extension and K un 2 its maximal unramified extension in a fixed algebraic closure of Q 2 . For an abelian variety A K 2 with potentially good reduction, there is a minimal extension M A K un 2 where A obtains good reduction. By a result of Serre-Tate ([43, §2, Corollary 3]), we have

M A = K un 2 (A[p]
) for any odd prime p. We recall that the curve E has potentially good reduction at 2. By Lemma 5.1, the same is true for J; so

M E = K un 2 (E[3]) and M J = K un 2 (J[3]). Proposition 5.4. We have M E = M J and Gal(M E K un 2 ) ≅ SL 2 (F 3 ).
Proof. Computing the standard invariants of E, we find that they have the following valuations at (2): (υ 2 (c 4 (E)), υ 2 (c 6 (E)), υ 2 (∆(E))) = [START_REF] Bosma | The Magma algebra system. I. The user language[END_REF][START_REF] Bosma | The Magma algebra system. I. The user language[END_REF][START_REF] Billerey | A multi-Frey approach to Fermat equations of signature (r, r, p)[END_REF].

Hence υ 2 (j(E)) = 11 and E has potentially good reduction at (2). It follows from [9, pp. 675, Corollaire] that E has semistability defect e = 24, hence Gal(M E K un 2 ) ≅ SL 2 (F 3 ) by [32]. On the other hand, J also has potentially good reduction at (2) by Lemma 5.1, and as a byproduct, the Dokchitser-Doris algorithm [START_REF] Dokchitser | Doris 3-torsion and conductor of genus 2 curves[END_REF] returns the totally ramified field K 2 (J[3]) with a defining polynomial over K 2 of degree 24.

We check that E has good reduction over

K 2 (J[3]), so M E ⊂ M J = K un 2 (J[3]) by minimality. Since [M E ∶ K un 2 ] = 24 = [M J ∶ K un 2 ]
the result follows.

The following well known result will be of use for us; due to a lack of a clear reference we include a proof here.

Lemma 5.5. Let K be a totally real field and q a prime above 2 in K. Let I q ⊂ G K be an inertia subgroup at q. Let h be a Hilbert modular form over K of level N and field of coefficients Q h . Assume that q N and that h has a supercuspidal exceptional type at q.

Then, for all primes p coprime to 6N and all primes P p in Q h , we have ρ h,P (I q ) ≃ SL 2 (F 3 ).

Proof. Let π h be the automorphic representation attached to h, and π h,q the local component at q. Also, let σ h,q ∶ W q → GL 2 (C) be the Weil representation attached to π h,q by the local Langlands correspondence (see [34]). Since π h,q is a supercuspidal exceptional representation, then σ h,q is an exceptional representation, which means that the projective image of σ h,q is either A 4 or S 4 (the A 5 case cannot occur since W q is solvable).

Let p be a rational prime coprime to 6N and P p a prime in Q h . Let D q ⊃ I q be a decomposition group at q in G K Theorem 5.6. Let P 7 in F = Q( √ 2) be a prime. Then, we have

M J = K un 2 (J[7]) = K un 2 (J[P]) and, moreover, ρ g,P I 2 ≅ ρ E,7 I 2 .
Here, I 2 denotes an inertia subgroup at (2) in G K .

Proof. Let P, P ′ be the two primes above 7 in Q( √ 2). By [46, Theorem 4.3.1], we have that

J[7] = J[P] × J[P ′ ].
This means that the field

M J = K un 2 (J[7]
) is the compositum of K un 2 (J[P]) and K un 2 (J[P ′ ]), the fields cut out by ρ J,P I 2 and ρ J,P ′ I 2 respectively.

We will now show that these three fields have the same degree and the first statement follows. By Theorem 5.3, we have ρ J,P ≅ ρ g,P and ρ J,P ′ ≅ ρ g,P ′ . By Proposition 5.4 and the discussion preceding it, we only need to show that the fields cut out by ρ g,P I 2 and ρ g,P ′ I 2 have degree # SL 2 (F 3 ) = 24.

Note that if ρ g,P I 2 is reducible then the conductor exponent at 2 is either 1 (special representation) or even (because the determinant of ρ g,P is cyclotomic, and hence on restriction to inertia, the diagonal characters must be inverses of each other); therefore, ρ g,P I 2 is irreducible and g has supercuspidal type at 2 which is not given by an induction from the unramified quadratic extension. We conclude that g is either supercuspidal induced from a ramified extension or exceptional. In the former case, then ρ g,P I 2 would have projective dihedral image; since the field cut out by ρ g,P I 2 is a Galois subextension of K un 2 (J[7]) K un 2 , which has Galois group SL 2 (F 3 ), and SL 2 (F 3 ) does not have any quotients which are dihedral, we conclude that g must have a supercuspidal exceptional type at 2. By Lemma 5.5, we obtain that the fields cut out by ρ g,P I 2 have degree # SL 2 (F 3 ) = 24 as required.

We now prove the last statement. From the first part of the theorem, Theorem 5.3 and Proposition 5.4, we have that M J is the field cut out by both ρ g,P I 2 and ρ E,7 I 2 . Thus, ρ g,P I 2 , and ρ E,7 I 2 have the same kernel, and image SL 2 (F 3 ) ↪ SL 2 (F 7 ) ⊂ GL 2 (F 7 ). Therefore, it follows from [26, Lemma 2] that they are isomorphic representations.

Finally, we now show Theorem 4.1 using the information on the inertial types. ] which asserts that ρ E a,b ,p ≅ ρ Z,p where Z is one of E 1,-1 , E 1,0 , E 1,1 . In the case p = 7, there is the additional possibility that ρ E a,b ,7 ≅ ρ g,P where P 7 in Q( √ 2). By [4, Remark 7.4] we have P = p 7 .

Suppose Z is one of the three curves above, the arguments of [4, Theorem 7] still hold for p = 7. For instance, if Z = E 1,-1 and 4 ∤ a + b, then it is shown that

(5.2) ρ E a,b ,p I 2 ≅ ρ Z,p I 2 .
To complete the proof, we only have to eliminate the possibility that ρ E a,b ,7 ≅ ρ g,p 7 .

By Theorem 5.3, we have ρ g,p 7 ≅ ρ J,p 7 . The proof of [4, Theorem 7 (A)] then remains valid by simply replacing Z K ′ with J K ′ w and using Lemma 5.2. This shows 13 a + b. Finally, we note that ρ g,p 7 I 2 ≅ ρ E 1,-1 ,7 I 2 by Theorem 5.6, and that ρ E 1,-1 ,7 I 2 ≅ ρ E a,b ,7 I 2 by (5.2), for 4 ∤ a + b. This shows that 4 a + b.

Remark 5.7. The obstruction to solving (1.1) comes from an abelian surface with real multiplication; namely, the surface A g attached to the form g. The approach in this section relies crucially on the fact that A g is isogenous over Q( √ 13) to a principally polarized abelian surface with real multiplication, i.e. A g is a Q( √ 13)-rational point on some Humbert surface. Hence, the methods in [17, §4.1.2] and [23, §7] can be applied to explicitly find a hyperelliptic curve C such that A g is isogenous over Q( √ 13) to the Jacobian J of C. There is thus a reasonable chance for the method in this section to succeed whenever the obstruction to the modular method for solving a Diophantine equation like the one in (1.1) is isogenous to a principally polarized abelian surface with real multiplication and reasonably-sized height.

Remark 5.8. Although we apply this generalized inertia argument to a situation with fixed exponent p = 7, the method described is also applicable to a setting with general exponent p (unlike the methods in Sections 6-7). This can be useful when working with certain Frey hyperelliptic curves as in the Darmon program for the Generalized Fermat equation [START_REF] Darmon | Rigid local systems, Hilbert modular forms, and Fermat's last theorem[END_REF].

For example, in forthcoming work [START_REF] Chen | A modular approach to Fermat equations of signature (p, p, 5) using Frey hyperelliptic curves[END_REF], the equation x p + y p = z 5 is studied using a Frey hyperelliptic curve C - 5 = C - 5 (a, b, c) constructed by Darmon (see [14, p. 425]), where (a, b, c) is a non-trivial primitive solution satisfying 2 ab, 5 ∤ ab. The Jacobian J -

5 = J - 5 (a, b, c) of C - 5 is defined over Q and becomes of GL 2 -type over F = Q( √ 5
) with real multiplication by F . In applying the modular method in this case, one finds that it is not possible to rule out an isomorphism (5.3) ρ J - 5 ,P ≅ ρ h,P by comparing traces of Frobenius at primes not dividing 10, where h is a certain Hilbert newform of parallel weight 2, trivial character, and level (2 4 r 2 ) over F ; here, P is a prime above p, r is the prime above 5 of F , and note that 2 is inert in F . By modularity and comparing a few traces, one checks that the Hilbert newform h corresponds to J - 5 (8, -8, 0) twisted by the quadratic character associated to F ( √ 2). Applying the Dokchitser-Doris algorithm [START_REF] Dokchitser | Doris 3-torsion and conductor of genus 2 curves[END_REF], we deduce that J - 5 (8, -8, 0) achieves good reduction over a degree 10 totally ramified extension of F un 2 (see [3]), so ρ h,P (I 2 ) has order dividing 20 and divisible by 5. Moreover, it is shown in [START_REF] Chen | A modular approach to Fermat equations of signature (p, p, 5) using Frey hyperelliptic curves[END_REF] that J - 5 (a, b, c) has potentially multiplicative reduction at 2 when 2 ab, 5 ∤ ab for p > 5. Therefore, this implies that ρ J - 5 ,P is a quadratic twist of Steinberg at 2 and hence ρ J - 5 ,P (I 2 ) has order dividing 2p. We thus obtain a contradiction to isomorphism (5.3) when p > 5.

A residual multiplicity one argument

In this section, we outline a proof of Theorem 4.1 based on establishing the mod 7 congruence mentioned in the introduction and described below; see also [START_REF] Billerey | A multi-Frey approach to Fermat equations of signature (r, r, p)[END_REF]Remark 7.4] for more details.

The natural attempt to use a Sturm bound such as the one in [START_REF] Gil | Hecke and Sturm bounds for Hilbert modular forms over real quadratic fields[END_REF] fails because it necessitates the computation of Hecke eigenvalues for primes of K = Q( √ 13) of norm at least 10 6 , which is not computationally feasible.

The method we introduce for establishing ρg,p 7 ≅ ρf,7 below, without the use of a Sturm bound, relies on a residual multiplicity one argument. It applies for more general f , g and p.

For example, it has also been used in [START_REF] Dembélé | An intriguing hyperelliptic Shimura curve quotient of genus 16[END_REF] to show the isomorphism of some residual mod 2 Galois representations arising from Hilbert modular forms over the maximal totally real subfield of Q(ζ 32 ), the cyclotomic field of 32nd root of unity.

Let E = E 1,-1 and p 7 be as in the previous section; in particular, we have F p 7 = F 7 .

Let S 2 (N) new be the new subspace of Hilbert cusp forms of weight 2, trivial character, and level N = (2 3 w 2 ). The elliptic curve E is modular by [27] and corresponds to a newform f ∈ S 2 (N) new . Similarly, in Theorem 5.3, we prove that the surface J is modular and corresponds to the form g ∈ S 2 (N) new . So, we have ρ f,7 ≃ ρ E,7 and ρ g,p 7 ≃ ρ J,p 7 . be the totally definite quaternion algebra over K ramified at both places at infinity only. Let O D be a maximal order in D. Also, let D = D ⊗ Ẑ, ÔK = O K ⊗ Ẑ, and ÔD = O D ⊗ Ẑ, where Ẑ is the profinite completion of Z. We fix an isomorphism ÔD × ≃ GL 2 ( ÔK ), and we define the compact open subgroup

U 0 (N) ∶= γ ∈ ÔD × ∶ γ ≡ * * 0 * mod N .
We consider the space S D 2 (N) = f ∶ D × D× U 0 (N) → C which has an action of Hecke operators T q for q ∤ N.

Let T N be the Hecke algebra acting on S D 2 (N) which is generated by the operators T q . There is a decomposition

S D 2 (N) = S D 2 (N) old ⊕ S D 2 (
S D 2 (N, A) ∶= S D 2 (N, Z) ⊗ A S D 2 (N, A) new ∶= S D 2 (N, Z) new ⊗ A.
We recall that, since [K ∶ Q] = 2 is even, the Jacquet-Langlands correspondence implies that there is an isomorphism of Hecke modules JL ∶ S D 2 (N) ≃ S 2 (N), which maps S D 2 (N) new onto S 2 (N) new . We let φ = JL -1 (f ) and ψ = JL -1 (g). Proposition 6.1. Consider the residual representations

ρg,p 7 ∶ Gal(Q K) → GL 2 (F 7 ) and ρf,7 ∶ Gal(Q K) → GL 2 (F 7 ).
Then, we have ρg,p 7 ≅ ρf,7 .

Proof. Recall that the coefficient field of f is Q and that of Up to scaling, we may assume φ and ψ are elements in S D 2 (N, O L,p 7 ) new and their reductions φ = θ(φ) and ψ = θ(ψ) are non-zero elements in S D 2 (N, F 7 ) new . Hence, φ and ψ are eigenvectors of T q (mod 7) for all q ∤ 7N.

g is L = Q( √ 2). Let O L = Z[ √ 2
Let T ⊆ End F 7 (S D 2 (N, F 7 ) new ) be the Hecke algebra generated by the operators T q (mod 7) for all primes q ∤ 7N, and Wφ (resp. W ψ) be the T-submodule generated by φ (resp. ψ) in S D 2 (N, F 7 ) new . Let T ′ ⊆ T be the subalgebra generated by the Hecke operators T q for prime ideals q ∤ 7N of norm up to 43, and S the socle of S D 2 (N, F 7 ) new considered as a T ′ -module, i.e. the sum of all simple T ′ -submodules of S D 2 (N, F 7 ) new , which is semi-simple. Since both φ and ψ are eigenvectors for T ′ , Wφ and W ψ are simple T ′ -modules of dimension one over F 7 . Hence, Wφ and W ψ are contained in S. They are isomorphic as T ′ -modules as for all prime ideals q ∤ 7N of norm up to 43, we have that a q (f ) (mod 7) = a q (g) (mod p 7 ) as elements in F 7 , where a q (u) denotes the eigenvalue of T q acting on an eigenvector u ∈ S D 2 (N). Using Magma [START_REF] Bosma | The Magma algebra system. I. The user language[END_REF], we can compute the T ′ -module S, which has dimension 348 over F 7 . There are 34 (non-isomorphic) simple constituents which have dimension one over F 7 , and each appears with multiplicity one. Thus,

Wφ = W ψ inside S ⊆ S D 2 (N, F 7 ) new . Since Wφ = W ψ inside S D
2 (N, F 7 ) new and both are T-modules as well, we obtain that a q (f ) (mod 7) = a q (g) (mod p 7 ) as elements of F 7 , for all prime ideals q ∤ 7N.

Second proof of Theorem 4.1. The bulk of the argument of the proof of this theorem given in Section 5, including the identity (5.2), still applies. Therefore, we only need to show that the isomorphism ρ E a,b ,7 ≅ ρ g,p 7 is not possible. Suppose we have that ρ E a,b ,7 ≅ ρ g,p 7 . Then by Proposition 6.1, we have ρ E a,b ,7 ≅ ρ g,p 7 ≅ ρE 1,-1 ,7 . Now the arguments that eliminated E 1,-1 apply to eliminate g, thereby completing the proof.

A unit sieve argument

In this section, we give a proof of Theorem 4.1 based on combining the study of units in classical descent with local restrictions on the solutions coming from the multi-Frey approach in [START_REF] Billerey | A multi-Frey approach to Fermat equations of signature (r, r, p)[END_REF].

Let ζ be a primitive 13th root of unity. Suppose (a, b, c) is a nontrivial primitive solution to (1.1). We then have the factorization in Z[ζ],

a 13 + b 13 = (a + b) 12 i=1 (a + ζ i b) = 3c 7 .
The integers a + b and a 13 +b 13 a+b are coprime away from 13 (see, e.g., [START_REF] Dahmen | Perfect powers expressible as sums of two fifth or seventh powers[END_REF]Lemma 2.1]). Let ≡ 1 (mod 13) be a prime number dividing a 13 +b 13 . Since a and b are coprime, we have that ∤ ab and hence there exists an integer b ′ such that bb ′ ≡ -1 (mod ). Therefore we have (ab ′ ) 13 ≡ 1 (mod ) and since ≡ 1 (mod 13), it follows that divides a + b. In particular, we have that 3 ∤ a + ζb. Furthermore, by classical descent, we have that (7.1)

a + ζb = β 7 if 13 ∤ a + b (1 -ζ)β 7 if 13 a + b,
where is a unit of Z[ζ] and β ∈ Z[ζ]. We only need to consider up to 7th powers, which means that there are initially 16, 807 possible choices for .

One can now reduce (7.1) modulo a prime q of Z[ζ] above the rational prime q. If q is such that the order of the multiplicative group of the residue field at q is divisible by 7, the condition of being a 7th power is nontrivial.

The primes q = 11, 17, 23, 29, 37, 41 satisfy the condition above; moreover, we can obtain local information from the modular method: if ρ E a,b ,7 ≅ ρ g,p 7 , locally at q this congruence imposes constraints on the solutions (a, b) modulo q (in both cases of good or multiplicative reduction of E a,b at q) and hence on the unit .

Note we cannot obtain information from q = 2 in the same way, because 2 divides the level of g. However, since the multiplicative group of the residual field of K at q = (2) has order a multiple of 7, this together with the assumption 4 ∤ a + b also imposes restrictions on .

Third proof of Theorem 4.1. We first note the condition 4 ∤ a + b is equivalent to 2 ∤ a + b for our equation. Now we sieve the set of possible units in the two cases of equation (7.1), assuming 2 ∤ a + b for a contradiction. More precisely, in both the cases 13 ∤ a + b and 13 a + b, using the primes q = 2, 11, 19, 23 the set of units which passes all local conditions is empty. Thus 4 a + b, as desired.

We will now prove that 13 a + b. From the above we can assume 4 a + b and 13 ∤ a + b for a contradiction. Using the local information at the primes q = 2, 11, 19, 23, 29, 41, again the set of units which passes all local conditions is empty, concluding the proof.

Remark 7.1. This method of sieving units also appears in [START_REF] Dahmen | Perfect powers expressible as sums of two fifth or seventh powers[END_REF], in the context of reducing the number of hyperelliptic curves to be considered for Chabauty. However, only the primes q not dividing the level are considered there. In our case, we succeed without the need to apply Chabauty since we can sieve also at q = 2 which is a prime in the level. It should also be noted that the reason the unit = 1 does not pass the local conditions from the modular method (unlike the situation in [START_REF] Dahmen | Perfect powers expressible as sums of two fifth or seventh powers[END_REF]) is because our original equation (1.1) does not have the trivial solutions ±(1, 0, 1), ±(0, 1, 1) due to the coefficient 3. To conclude this paper, we use the technique from Section 3 to complete the following result of Anni-Siksek [START_REF] Anni | Modular elliptic curves over real abelian fields and the generalized Fermat equation x 2 + y 2m = z p[END_REF].

Theorem 8.1 (Anni-Siksek). Let p = 3, 5, 7, 11 or 13. Let , m ≥ 5 be primes, and if p = 13 suppose moreover that m, ≠ 7. Then the only primitive solutions to x 2 + y 2m = z p are the trivial ones (x, y, z) = (±1, 0, 1) and (0, ±1, 1).

The proof of this theorem is a remarkable application of the modular method over totally real subfields of Q(ζ p ) + . The extra conditions m, ≠ 7 for p = 13 are required due to the presence of a single Hilbert newform, denoted f 11 in loc. cit, which evades the elimination step.

Let K ⊂ Q(ζ 13 ) + be the cubic field of Section 3. Recall that 2 is inert in K and that q 13 is the unique prime above 13. The newform f 11 belongs to S 2 (2 ⋅ q 13 ) new , the new subspace of Hilbert cusp forms of level 2 ⋅ q 13 , weight 2, and trivial character. It is the unique newform in this space with field of coefficients Q f 11 = Q(ζ 7 ) + the real cubic subfield of the cyclotomic field of 7-th roots of unity, in which 7 is totally ramified. Let p 0 be the unique prime of Q f 11 above 7. The authors are not able to exclude the possibility that (8. [START_REF] Anni | Modular elliptic curves over real abelian fields and the generalized Fermat equation x 2 + y 2m = z p[END_REF] ρ E ′ ,7 ≃ ρ f 11 ,p 0 , where E ′ is the Frey curve defined in [1, pg. 10]; see [1, pg. 19] for further details. However, as observed in loc. cit., numerical evidence strongly suggests that ρ ss f 11 ,p 0 ≃ χ 7 ⊕ 1 so, in particular, ρ f 11 ,p 0 is reducible. In Proposition 8.3, we show that this is indeed the case. This means that the isomorphism (8.1) cannot occur since ρ E ′ ,7 is irreducible. Therefore, since p 0 is the only prime in Q f 11 above 7, we have successfully completed the elimination step for the form f 11 . This in turn allows us to remove the assumption that m, ≠ 7 for p = 13 in Theorem 8.1. Thus we have the following corollary.

Corollary 8.2. Let , m ≥ 5 be primes. Then the only primitive solutions to x 2 + y 2m = z 13 are the trivial ones (x, y, z) = (±1, 0, 1) and (0, ±1, 1).

We now complete the proof of this result by showing that ρ f 11 ,p 0 is indeed reducible.

Proposition 8.3. The representation ρ f 11 ,p 0 ∶ G K → GL 2 (F 7 ) is reducible.

Proof. Suppose ρ f 11 ,p 0 is irreducible for a contradiction.

The conductor of ρ f 11 ,p 0 is N ′ = (1), (2), q 13 or 2q 13 . From the irreducibility assumption on ρ f 11 ,p 0 and [6, Theorem 3.2.2] ,we conclude there exists a Hilbert newform g ∈ S 2 (N ′ ) new such that ρ f 11 ,p 0 ≃ ρ g,p ′ for some prime p ′ 7 in the coefficient field Q g of g.

For N ′ = (1) and N ′ = q 13 , we have dim S 2 (N ′ ) new = 0, so N ′ = (2) or 2q 13 .

For N ′ = (2), we have dim S 2 (N ′ ) new = 1. The unique newform corresponds to the isogeny class of the base change to K of the elliptic curve W with Cremona label 338b1. From LMFDB [36, Elliptic Curves over Q], we see that ρ W,7 is reducible, hence ρ f 11 ,p 0 ≃ ρ W,7 G K .

We conclude that ρ f 11 ,p 0 ramifies at both (2) and q 13 , that is N ′ = 2q 13 . Now, a similar argument to the one used in the proof of Proposition 3.2 shows that ρ f 11 ,p 0 extends to an irreducible representation ρ ∶ G Q → GL 2 (F 7 ), and that there is a classical newform h ∈ S 2 (26) new such that ρ ≃ ρ h,p for some prime p 7 in Q h . There are two newforms h 1 , h 2 ∈ S 2 (26) new , with rational coefficients, corresponding to the isogeny classes of the elliptic curves W 1 and W 2 with Cremona label 26a1 and 26b1 respectively. The curve W 2 has a 7-torsion point, so ρ h 2 ,7 is reducible, hence ρ ≃ ρ h 2 ,7 .

Finally, we observe that a 5 (W 1 ) = -3 for the prime 5, which is split in K. So, for a prime q 5 in K, we have Tr(ρ h 1 ,7 G K (Frob q )) = Tr(ρ W 1 ,7 (Frob 5 )) ≡ -3 (mod 7).

But, we easily check that Tr(ρ f 11 ,p 0 (Frob q )) ≡ 6 ≡ -3 (mod p 0 ), showing that ρ ≃ ρ h 1 ,7 . Since there are no other possible forms h we obtain the desired contradiction.

Theorem 4 . 1 .

 41 Let (a, b, c) be a non-trivial primitive solution to equation (1.1) for p = 7. Then 13 a + b and 4 a + b.

From [ 4 ,

 4 Remark 7.4], Theorem 4.1 follows provided we can discard the form g. The following theorem reduces the resolution of (1.1) for p = 7 to the elimination of the form g by discarding the other four obstructing forms. Theorem 4.2. Assume Theorem 4.1. Then Theorem 1.1 holds. Proof of Theorem 4.2. It suffices to consider x 13 + y 13 = 3z n with n = p a prime number. The case p = 7 is precisely [4, Theorem 2], so we can assume p = 7. Suppose (a, b, c) is a non-trivial primitive solution to equation (1.1) with p = 7. From Theorem 4.1 we can assume that 4 a + b and 13 a + b.

First proof of Theorem 4 . 1 .

 41 Let (a, b, c) be a non-trivial primitive solution to equation (1.1) with n = p = 7. Let E a,b be the Frey curve defined in [4, Section 7.1]. The proof of [4, Theorem 7] uses [4, Proposition 9

  N) new .where the subspaces S D 2 (N) old and S D 2 (N) new are T N -stable, andS D 2 (N) new is the orthogonal complement of S D 2 (N) old under a certain inner product defined on S D 2 (N). The Z-submodule of S D 2 (N) given by S D 2 (N, Z) = f ∶ D × D× U 0 (N) → Z , is stable under T Nand generates S D 2 (N) over C (i.e. is an integral structure for S D 2 (N)). Let S D 2 (N, Z) new be the orthogonal projection of S D 2 (N, Z) to S D 2 (N) new . Then S D 2 (N, Z) new is an integral structure for S D 2 (N) new ; in particular, the matrices representing the action of the T q on S D 2 (N) new have integer coefficients. For any commutative ring with unity A, define

  ] and O L,p 7 be the completion of O L at p 7 . Letθ ∶ S D 2 (N, O L,p 7 ) new ↠ S D 2 (N, F 7) new be the natural reduction map.

  8. A remark on the equationx 2 + y 2m = z p

  [START_REF] Elkies | K3 surfaces and equations for Hilbert modular surfaces[END_REF] Theorem 17], there is an equation for the Humbert surface H 8 of discriminant 8, which parametrizes principally polarized abelian surfaces with RM by Z[ √ 2] (but where the action of Z[ √ 2] by endomorphisms is forgotten), and is birational to the projective plane P 2 r,s . Let Y -(8) be the Hilbert modular surface in[START_REF] Elkies | K3 surfaces and equations for Hilbert modular surfaces[END_REF] which is the coarse moduli space which parametrizes principally polarized abelian surfaces with real multiplication by Z[ √ 2]. The loc. cit. gives the Hilbert modular surface Y -(8) as a double-cover of P 2 r,s and this is a birational model of Y -(8) over Q. Additionally, a birational map from Y -(8) to the moduli space M 2 of genus 2 curves is described. When this map is defined and evaluated at a point (A, φ), where φ is the moduli structure imposed on A by Y -(8), it gives the Igusa-Clebsch invariants of the genus 2 curve C ′ whose Jacobian is A.

	determines the K-isomorphism class of an abelian surface A arising as the Jacobian of C ′
	whose Igusa-Clebsch invariants are given by
	I ′ 2 =	18112u -38832 81	,
	I ′ 4 =	-112736u + 270660 6561	,
	I ′ 6 =	2386589920u -5484934104 531441	,
	I ′ 10 =	532320256u -1222121472 3486784401	.
	The point
	(r ′ , s ′ ) ∶=	-20u -24 81	,	-1456u + 3354 81

We acknowledge the financial support of ANR-14-CE-25-0015 Gardio (N. B.), an NSERC Discovery Grant (I. C.), and the grant Proyecto RSME-FBBVA 2015 José Luis Rubio de Francia (N. F.).

Université Clermont Auvergne, CNRS, LMBP, F-63000 Clermont-Ferrand, France.


Philippe Maugis
email: philippe.maugis@univ-amu.fr

Thermo-kinetic modelling of the giant Snoek e ect in carbon-supersaturated iron

Keywords:

Snoek relaxation in interstitial bcc solid solutions is the origin of the Snoek peak in frequencydependent and temperature-dependent internal friction measurements. When applied to low-carbon steels, the internal friction profiles show a linear dependence of the peak height with carbon content. However, recent Monte Carlo simulations of frequency-dependent internal friction exhibited a nonlinear giant peak height at high carbon contents. To investigate this e ect in temperature-dependent internal friction, we developed a thermo-kinetic mean-field theory of the Snoek relaxation phenomenon. By taking into account the collective behaviour of the interstitial atoms, our theory predicts (i) a nonlinear dependence of the peak height with composition when approaching the order-disorder transition of the alloy; (ii) a shift in the temperature of the peak; (iii) a composition-dependent activation enthalpy of the relaxation time. The theory is exemplified by the case of carbon-supersaturated iron.

Introduction

The Snoek e ect in body-centred solid solutions originates from the tendency of interstitial atoms to occupy specific non-equivalent sites in strained crystals: the elastic strain field energetically favours one of the three sets of interstitial sites, inducing a redistribution of the interstitial atoms via atomic jumps towards the favoured sites. Now, as each interstitial deforms the host matrix, the homogeneous strain of the crystal results not only from the applied stress but also from the distribution of the interstitial atoms over the atomic sites. This strain in turn interacts with the interstitial atoms and a ects the energetic levels of both stable and transition sites. The thermodynamics of the solid solution as well as the kinetics of the atomic jumps are thus a ected by both the applied stress and the interstitial distribution.

In 1948, Zener [START_REF] Zener | Theory of strain interaction of solute atoms[END_REF] showed that the alloy thermodynamics based on carbon-strain interaction in a stress-free Fe-C crystal leads to carbon long-range ordering below a transition temperature. By a mean-field model, Shtremel [START_REF] Shtremel | Influence of stresses on order in interstitial solutions[END_REF] showed that an applied stress can favour Zener ordering. These ideas have been confirmed since by various theoretical approaches [START_REF] Khachaturyan | On the theory of the ordering of carbon atoms in a martensite crytal[END_REF][START_REF] Taylor | Ageing of ferrous martensites[END_REF][START_REF] Chirkov | Tetragonality and the distribution of carbon atoms in the Fe-C martensite: Molecular-dynamics simulation[END_REF][START_REF] Maugis | Temperature hysteresis of the order-disorder transition in carbonsupersaturated ↵-Fe[END_REF][START_REF] Kurdjumov | Nature of axial ratio anomalies of the martensite lattice and mechanism of di usionless gamma to alpha transformation[END_REF][START_REF] Fan | Lattice-parameter variation with carbon content of martensite. II. Long-wavelength theory of the cubic-to-tetragonal transition[END_REF][START_REF] Udyansky | Interplay between long-range elastic and short-range chemical interactions in Fe-C martensite formation[END_REF][START_REF] Udyansky | Orientational ordering of interstitial atoms and martensite formation in dilute Febased solid solutions[END_REF][START_REF] Zhang | Mechanism of collective interstitial ordering in Fe-C alloys[END_REF][START_REF] Naraghi | Stability of Fe-C Martensite-E ect of Zener-Ordering[END_REF][START_REF] Naraghi | Thermodynamics of stable and metastable structures in Fe-C system[END_REF][START_REF] Sinclair | Molecular dynamics study of the ordering of carbon in highly supersaturated alpha-Fe[END_REF][START_REF] Chirkov | Molecular-dynamics Simulations of Carbon Ordering in bcc Fe and its Impact on Martensite Transition[END_REF][START_REF] Waseda | Ordering of carbon in highly supersaturated ↵ -Fe[END_REF][START_REF] Maugis | Ferrite, martensite and supercritical iron: A coherent elastochemical theory of stress-induced carbon ordering in steel[END_REF][START_REF] Maugis | Stress-controlled carbon diffusion channeling in bct-iron: A mean-field theory[END_REF][START_REF] Maugis | Stability of Zener order in martensite: an atomistic evidence[END_REF]. The e ect can be understood as follows: an applied stress creates a strain (the elastic response) that favours one set of octahedral sites. These sites becoming progressively over-populated create an additional strain (the anelastic response) that in turn favours the sites even more. As a result, when applied to a low-carbon alloy, the stress induces a low-order state, the ferrite phase; above a threshold carbon content, it produces a high-order state, the martensite phase. In the vicinity of the threshold composition, the collective response of the carbon atoms produces a very large anelastic strain response, which is the origin of the giant Snoek e ect [START_REF] Zener | Elasticity and anelasticity of metals[END_REF][START_REF] Maugis | Giant Snoek peak in ferrite due to carbon-carbon strain interactions[END_REF].

The carbon-carbon long-range elastic interaction therefore modifies the Snoek relaxation behaviour. Several theoretical works have focused on this point. The coupling model [START_REF] Dattagupta | Strain ordering in BCC metals and the associated anelasticity[END_REF][START_REF] Ghoshal | Dattagupta, 3-d Model for strain ordering in steel: II relaxational e ects[END_REF][START_REF] Ngai | Theoretical basis and general applicability of the coupling model to relaxations in coupled systems[END_REF] is derived from a mean-field approximation of the elasticity theory of point defects. The random cooperative strain interaction model (RCSI) [START_REF] Haneczok | Internal Friction Studies on Oxygen-Oxygen Interaction in Niobium[END_REF] assumes a random distribution of the strain interactions. These models predict a shift and a broadening of the internal friction profiles. However, they do not consider the consequence of long-range carbon-carbon interaction on the saddle point energy of the jumping atoms. Recently, the author showed by Monte Carlo simulations that this interaction has a strong e ect on the relaxation time [START_REF] Maugis | Giant Snoek peak in ferrite due to carbon-carbon strain interactions[END_REF].

In this paper, we present a mean-field approach of the Snoek relaxation phenomenon. Our theory is based on the elastochemical model describing the thermodynamics of the interstitial solid solution [START_REF] Maugis | Nonlinear elastic behavior of iron-carbon alloys at the nanoscale[END_REF], and on the rate theory applied to carbon jumps [START_REF] Maugis | Stress-controlled carbon diffusion channeling in bct-iron: A mean-field theory[END_REF][START_REF] Lawrence | Carbon di usion in supersaturated ferrite : A comparison of mean-field and atomistic predictions[END_REF][START_REF] Maugis | Revisiting the pressure e ect on Carbon migration in iron[END_REF]. Both thermodynamic and kinetic aspects rely on the elasticity theory of point defects to describe the long-range carbon-carbon interaction. This thermo-kinetic model is applied to internal friction measurements. Our results are compared to Monte Carlo simulations of frequency-dependent profiles and to an experimental temperature-dependent profile. The focus is then set on the e ect of carbon content on the internal friction. The nonlinear peak height-carbon content relationship and the giant Snoek e ect in supersaturated alloys are highlighted. The limits of the standard linear-response approximation [START_REF] Nowick | Anelastic relaxation in crystalline solids[END_REF] are also discussed.

The thermo-kinetic model

The thermo-kinetic model is a consistent description of the thermodynamics of carbon long-range ordering and of the dynamics of carbon atomic jumps. As was demonstrated in previous papers [START_REF] Maugis | Temperature hysteresis of the order-disorder transition in carbonsupersaturated ↵-Fe[END_REF][START_REF] Maugis | Stability of Zener order in martensite: an atomistic evidence[END_REF], Zener ordering mostly depends on the long-range elastic carbon-carbon interactions. Therefore, short-range interactions can be neglected. The mean-field approximation implies the use of average site fractions of carbon C i on each sublattice i, and the consideration of the homogeneous strain ", irrespective of the strain heterogeneities. The core of the model is the mathematical expression of the enthalpy of the crystal as function of composition, temperature and applied stress. From the enthalpy function, the migration enthalpy of the carbon atoms is expressed, leading to the rate equations of the site fractions, and by integration, to the time-dependent strain response. The main features of the model are described below.

Equations of the model

Carbon atoms soluted in the bcc lattice sit on the stable octahedral sites. They migrate via the unstable tetrahedral sites. There are 3 non-equivalent octahedral configurations (labelled i = 1, 2, 3) and 3 non-equivalent tetrahedral configurations (labelled j = 1, 2, 3). Each configuration is characterised by its force dipole tensor P O i or P T j representing the elastic field induced by the defect. In our mean-field approach, each carbon atom interacts with the elastic field created by both the applied stress and the other carbon atoms. The resulting homogeneous strain tensor " is expressed as a function of the stress tensor and of the force dipole density tensor p [START_REF] Maugis | Nonlinear elastic behavior of iron-carbon alloys at the nanoscale[END_REF]:

" = S ( + p) . ( 1
)
S is the elastic compliance tensor of the crystal. The dipole density p depends on the distribution of the carbon atoms over the sites. When all carbon atoms lie on octahedral sites, p is written as

p = 1 V 0 … C i P O i , ( 2
)
where V 0 is the atomic volume of the lattice. The dipole density of the crystal varies during and after each carbon jump. For a given distribution of the carbon atoms over the interstitial sites, the elastic contribution to the total enthalpy is then written, for a crystal of volume V ,

H = * 1 2 V S( + p) ( + p). ( 3
)
As an equivalent to the force dipoles tensors, we introduce the strain dipole tensors = SP _V 0 . According to the tetragonal symmetry of the defects, each of these tensors has a singulet component 1 representing the strain induced by a carbon atom along the tetragonal axis, and a doublet component 2 representing the strain in perpendicular directions. With these notations, and making use of Equation 3, the migration enthalpy of a carbon atom from site i through the transition site j is written [START_REF] Maugis | Ferrite, martensite and supercritical iron: A coherent elastochemical theory of stress-induced carbon ordering in steel[END_REF]

H m iôj = H m 0 * V 0 T j * O i ( + p), (4)
where H m 0 is the migration enthalpy in the stress-free carbonfree crystal. It is clear from this equation that not only the applied stress but also the carbon distribution a ect the migration enthalpy.

According to the jump rate theory, the probability per second for a carbon atom to jump from stable site i through the transition site j is written

p iôj = ⌫ 0 exp ⇠ *H m iôj _k B T ⇡ . ( 5
)
From this probability function, too means of time integration are available. The first one is the Monte Carlo method. On-lattice atomic kinetic Monte Carlo (AKMC) was applied successfully to study the thermodynamic [START_REF] Maugis | Ferrite, martensite and supercritical iron: A coherent elastochemical theory of stress-induced carbon ordering in steel[END_REF], thermo-elastic [START_REF] Maugis | Nonlinear elastic behavior of iron-carbon alloys at the nanoscale[END_REF], di usional [START_REF] Maugis | Stress-controlled carbon diffusion channeling in bct-iron: A mean-field theory[END_REF] and relaxation [START_REF] Maugis | Giant Snoek peak in ferrite due to carbon-carbon strain interactions[END_REF] properties of the Fe-C system. Details of the method can be found in reference [START_REF] Maugis | Ferrite, martensite and supercritical iron: A coherent elastochemical theory of stress-induced carbon ordering in steel[END_REF].

The second means of time integration is the mean-field approach used in the thermo-kinetic model. In this approach, the jump frequency from i through j is written iôj = C i p iôj . The net flux from stable position i to stable position k via the saddle position j is then J iôk = iôj * kôj . An octahedral site of type i has two nearest neighbours of type k and two nearest neighbours of type k ® . Then the matter balance at site i implies the rate equations

Ü C i = 2 J kôi + J k ® ôi , i= 1, 2, 3. (6)
These equations are conservative, such that the total carbon fraction C = ≥ C i remains constant. When integrated, the coupled equations ( 6) provide the time evolution of the site fractions C i (t) under given applied stress. The resulting strain response is provided by Equation 1.

In Section 3, the thermo-kinetic model is compared to Monte Carlo simulations.

The linear-response approximation

When the applied stress is low, the strain response is also low, unless certain circumstances discussed later. Then the strain response is accurately approximated by a linear function of the stress. Similarly to reference [START_REF] Maugis | Giant Snoek peak in ferrite due to carbon-carbon strain interactions[END_REF], we used this approximation to model the internal friction in a forced torsion pendulum experiment. We considered a crystal submitted to a pure (001)[110] shear stress such that 22 = * 11 = (t). The crystal response to a sinusoidal excitation of pulsation ! is a sinusoidal pure shear strain "(t) = " 22 * " 11 . The time lag between stress and strain corresponds to the energy loss Q *1 , which follows a Debye curve [START_REF] Nowick | Anelastic relaxation in crystalline solids[END_REF]:

Q *1 = !⌧ 1 + (!⌧) 2 (7)
From our thermo-kinetic model, we established the equations relating the relaxation strength and the relaxation time ⌧ to the carbon content [START_REF] Maugis | Giant Snoek peak in ferrite due to carbon-carbon strain interactions[END_REF]. The calculated relaxation strength is a non-linear function, exhibiting a singularity at C = C * :

(C) = C C * * C (8)
Carbon fraction C * is the value above which the solid solution becomes unstable relatively to Zener ordering. It depends linearly on temperature as

C * = 3S ® 2V 0 ( 1 * 2 ) 2 k B T . ( 9
)
This equation defines a spinodal line in the temperaturecomposition phase diagram. We introduced the shear compliance S ® = 2(S 11 * S 12 ). The compression compliance will also be of use:

S ®® = S 11 + 2S 12 .
The relaxation time exhibits the same singularity as the relaxation strength:

⌧(C) = (6⌫ 0 ) *1 exp L H m 0 + E HZ C k B T M C * C * * C ( 10
)
where ⌫ 0 is the attempt frequency for carbon migration and H m 0 is the migration enthalpy in the infinite dilution limit. The novelty of our approach is the expression of the compositiondependent activation enthalpy of the relaxation time:

H m = H m 0 + E HZ C, (11)
with

E HZ = * V 0 3S ®® tr O tr T * O . ( 12
)
The energy coe cient E HZ depends only on the elastic properties of the material. It originates from the e ect of strain (both volumic and shear components) on the migration enthalpies of the interstitials [START_REF] Maugis | Stress-controlled carbon diffusion channeling in bct-iron: A mean-field theory[END_REF]. Notice that in the limit of low carbon content (C ô 0) one recovers from Equation 10the usual expression of the Snoek relaxation time

⌧ *1 = 6⌫ 0 exp *H m 0 _k B T .
The consequences of the composition dependence of and ⌧ are (i) the giant Snoek relaxation in the vicinity of the ordering transition and (ii) the shift and broadening of the Snoek peak in frequency-dependent internal friction [START_REF] Maugis | Giant Snoek peak in ferrite due to carbon-carbon strain interactions[END_REF]. Such e ects are investigated further in Section 3.

In order to explore the influence of temperature, we can alternatively express the relaxation strength and time as functions of T . Actually, by substituting C * (Eq. 9) into Eqs. 8 and 10, the relaxation strength is written

(T ) = T * T * T * (13)
and the relaxation time

⌧(T ) = (6⌫ 0 ) *1 exp L H m 0 + E m C C k B T M T T * T * (14)
with temperature T * defined by

k B T * = 2V 0 ( 1 * 2 ) 2 3S ® C. ( 15
)
In the vicinity of T = T * the relaxation strength (T ) diverges with the power law (T * T * ) *1 . However, this produces no divergence of the energy loss because the relaxation time ⌧(T ) diverges with the same law. Hence, in the vicinity of T * the increase in relaxation strength is canceled out by the increase in relaxation time. As a result, the linearresponse approximation predicts no discontinuity in energy loss when crossing T * :

Q *1 (T * ) = 6⌫ 0 ! exp 4 S ® 2S ®® tr( O )tr( T * O ) ( 1 * 2 ) 2 5 ù exp L * H m 0 k B T * M . ( 16
)
Such a property was previously described by Nowick and Berry [START_REF] Nowick | Anelastic relaxation in crystalline solids[END_REF]. It will be further discussed in Section 4.2.

Model parameters and computation set up

We assumed that the material parameters do not depend on temperature in the range of interest (250-400K). We used the lattice parameter, elastic sti ness tensor and force dipole tensors computed from first principles in reference [START_REF] Maugis | Stress-controlled carbon diffusion channeling in bct-iron: A mean-field theory[END_REF], such that they are consistent with one another. The values are reported in Table 1, together with the symmetrised compliances and the strain dipole tensors deduced from the elasticity theory. The values are compatible with the data found in the literature. The attempt frequency and migration enthalpy for carbon were fitted to the data of da Silva [START_REF] Silva | Di usion of carbon and nitrogen in B.C.C. iron[END_REF]: ⌫ 0 = 159 THz and H m 0 = 0.872 eV. The phase diagrams were constructed by minimising the Gibbs energy function G = H * T S. H is the enthalpy function of Equation 3 and the S is the configurational regular entropy.

Snoek relaxation profiles were obtained from successive computations of the thermo-kinetic model at selected frequencies f = !_2⇡. High-accuracy Runge-Kutta integration of the rate equations was performed to get the timedependent site fractions. The internal friction is defined as the fractional energy loss per cycle:

Q *1 = 1 2⇡ W W . ( 17
)
W is the maximum elastically stored energy per unit volume during a cycle. W was computed as the integral of d" along a cycle. When varying the temperature for internal friction computations, the starting state of the alloy was set to fully disordered. In doing so, we simulated internal friction measurements upon cooling of the ferrite phase.

Comparison with experiment

Q *1 poly = 2S ® 5G *1 Q *1 single ( 18
)
where G *1 = (3_5)S 44 + (2_5)S ® is the average inverse torsion modulus. With the data of Table 1, we have

Q *1 poly = 0.516 Q *1
single . Comparison with experiment is satisfactory, given that no fitting parameter was used (Fig. 1) Table 1 Top rows: lattice parameter a 0 (in nm), elastic compliances S ij (in GPa *1 ) and dipole moments P a and P c (in eV) computed by DFT. Bottom rows: symmetrised compliances (in GPa *1 ) and dipole strains deduced from elasticity theory.

a

Temperature T [K]

Experiment Thermo-kinetic model

Frequency-dependent internal friction

Frequency-dependent internal friction is usually measured at room temperature in a series of runs at various oscillation frequencies. Figure 2 presents a set of Snoek profiles computed with the thermo-kinetic model at temperature T = 300 K over a wide range of oscillation frequency. Specific carbon contents have been selected: C = 0.5% is just beyond the limit of the composition-peak height linear relationship; C 0 = 1.225% coincides with the Zener order-disorder transition; and C = 1.37% is close to the spinodal composition of C * = 1.379% (Eq. 9). We see in Figure 2 that the peak height increases rapidly with increasing carbon content, while the peak position shifts towards lower frequencies. The profiles also appear broadened and skewed at high carbon content.

These results are compared to kinetic Monte Carlo simulations reported in reference [START_REF] Maugis | Giant Snoek peak in ferrite due to carbon-carbon strain interactions[END_REF]. The simulations used an applied stress amplitude of 100 MPa, a high value as reference to experimental usage, but technically necessary to overcome the high fluctuations present in the finite-size simulation box. The coincidence between both approaches is very good, which validates the thermo-kinetic model. The later has the advantage of being far less computationally demanding than the Monte Carlo simulations.

The steep increase in peak height when C approaches Figure 3, top gathers the stress-strain hysteresis cycles corresponding to the peak frequencies of Figure 2. The cycle at C = 0.5% has an ellipsoid shape, as expected in the linear regime. When the carbon content increases, the cycle grows and its largest diagonal declines. This decline corresponds to an e ective softening of the alloy. Correlatively, the cycle area increases largely, in direct relation with the increase in energy loss.

The very large energy loss when C approaches C * can be understood from the quasi-static stress-strain isotherms drawn in Figure 3, bottom. These curves have been computed with a very small stress rate insuring equilibrium at each time step, starting from the disordered state at = 0. Close to the linear regime (C = 0.5%), the stress-strain characteristic is almost a straight line, whose slope is slightly less than the inverse shear compliance 1_S ® . When the carbon content increases, the characteristic becomes non linear and the slope at low stress continues to decrease. At C = 1.37%, the curve is almost flat in the strain range of [*0.79%, 0.79%] under low applied stress, and steep beyond. Such a superelastic behaviour is due to the collective migration of the carbon atoms towards the favoured interstitial sites. The carbon atoms induce a large anelastic strain at little stress expense (0.74 MPa). This arises because the disorder to order transition becomes barrier-less when the spinodal line is reached. Consequently, the static compliance is almost infinite. Under dynamic conditions, the corresponding energy loss is naturally very large, as seen in Figure 2.

Temperature-dependent internal friction

Temperature-dependent internal friction is obtained experimentally by a series energy-loss measurements performed at various temperatures. The resulting profiles depend on the oscillation frequency. We will show that they are strongly carbon-dependent. The results will be discussed in relation to the thermo-elastic properties of the Fe-C system.

The thermo-elastic properties of Fe-C

During a temperature-dependant internal friction measurement, the thermo-elastic properties of the material are scanned over a range of temperature. Zener ordering may occur in that range, to which the Snoek relaxation is sensitive. In the present work, we focused on carbon-supersaturated alloys, i.e. carbon is assumed to remain in solid solution. This can be achieved in practice by measuring the Snoek profile during cooling from a not too elevated temperature. Under this condition, the thermo-elastic properties of the alloy are expected to evolve strongly in the vicinity of the ferrite to martensite transition temperature [START_REF] Maugis | A Temperature-Stress Phase Diagram of Carbon-Supersaturated bcc-Iron, Exhibiting "Beyond-Zener" Ordering[END_REF].

To investigate this point and shed light on the giant relaxation phenomenon in temperature-dependant internal friction measurements, the temperature-stress phase diagram and the stress-strain isotherms we computed in the case of a carbon content C = 1.37 at.% (0.30 wt.%).

The stress-composition phase diagram drawn in Figure 4 was constructed by minimisation of the Gibbs energy. The first-order transition lines form a symmetric Y-shape. At high temperature, the ferrite phase is stable. It is characterised by a low degree of Zener order (total disorder is achieved at zero stress). The martensite phase is stable at low temperature. It has a high degree of Zener order, i.e. most carbon atoms sit on one type of octahedral sites. Zener orderdisorder transitions occur along the inclined transition lines, which join at zero-stress and temperature T 0 = 322 K. K and K ® are two critical points beyond which the order-disorder transition is continuous ( K ® = * K = 9.04 MPa and T K = T K ® = 335 K). An important point for the thermo-elastic behaviour of the alloy is the spinodal point at zero stress, located at temperature T * = 298 K.

When the temperature is decreased, under zero-stress condition, the ordering transition can occur below T 0 . However, disorder may persist as a metastable state until temperature T * (Eq. 15) is reached, at which stage the ordering transition occurs barrier-less. The stress-composition phase diagram is homothetic in carbon content, i.e. a diagram at composition C can be deduced from Figure 4 by rescaling both axes proportionally to C.

The thermo-elastic behaviour is related to the temperaturestress phase diagram. Selected stress-strain isotherms of the alloy C = 1.37 at.% are gathered in Figure 4, bottom. Notice that the stress axis is horizontal, and coincides with the phase diagram on top. At T = 400 K, a temperature far above the ordering transition, the stress-strain characteristic is almost linear, with a slope slightly higher than the shear compliance S ® . When the temperature is decreased, strong non-linearities appear, together with an increase in slope at the origin. At the temperature of the critical points, T K , the slope is infinite when the stress equals the critical value of ±9.04 MPa corresponding to the critical points K and K ® . Below T K , a step in strain accompanies the order-disorder transition when crossing the K ® T 0 K line. The amplitude of the step is maximum at the spinodal temperature (T * ). At this temperature, a vanishing stress produces a finite strain, i.e. the static compliance in infinite, as expected at a critical point [START_REF] Zener | Theory of strain interaction of solute atoms[END_REF][START_REF] Landau | Statistical physics[END_REF]. Shear strain ε

[%] Shear stress σ [MPa] 298 K (T -) 322 K (T 0 ) 335 K (T K ) 360 K 380 K 400 K

Influence of the oscillation frequency

The computer implementation of the thermo-kinetic model allows for fast computation of low-frequency and low-stress simulations. The stress amplitude was set to 1 MPa in a series of computations where the oscillation frequency f was varied from 10 *2 to 100 Hz. We obtained the profiles presented in Figure 5.

As expected for thermo-activated processes, the peak position shifts toward low temperatures when decreasing the frequency, while the peak height increases. The high-frequency profiles (f OE 1 Hz) are well reproduced by the linear-response approximation. In this regime, the system is driven out-ofequilibrium at a frequency that does not allow enough time for the ordering transition to take place, although the temperature is brought below T 0 . This is not the case at lower frequencies (f Õ 10 *1 Hz). In particular, at f = 10 *2 Hz the sharp drop in energy loss located at 307 K is caused by the ferrite to martensite transition. Such a transition is a nonlinear e ect, out of reach of the linear model [START_REF] Nowick | Anelastic relaxation in crystalline solids[END_REF].

E ect of carbon content

The e ect of carbon content was studied at the usual oscillation frequency of 1 Hz. The linear model proved to be a good approximation at this frequency. It was run for carbon contents up to 1.37 at.% (Fig. 6). Two important features emerge. First, the peak height is a steep function of carbon content in the vicinity of 1.4 at.%, as can be seen in Figure 7. The peak-height-carbon content relationship is linear only for carbon content as low as Ì 0.2 at.%, similarly to the frequency-dependent case (see Section 3).

Second, the peak position is composition-dependent: it is equal to 310 K at low carbon but shifts towards higher temperatures when the carbon content increases, up to 334 K when C = 1.37% (Fig. 7). Given the temperature dependence of both (T ) and ⌧(T ) (Eqs. 13, 14 and Fig. 8), the peak coordinates are not analytical. The shift can neverthe-less be qualitatively understood from the increase in activation enthalpy H m with carbon content (Eq. 11): numerically H m = 0.872+0.600C, which varies from 0.872 to 0.880 eV in the range of C = 0 to C = 1.37%.

Discussion

The thermo-kinetic model predicts that the peak energy loss in internal friction measurements should reach huge values when the carbon content is close to the Zener orderdisorder transition. This giant Snoek relaxation is non-linear with respect to the carbon content. It is accompanied by a shift in the peak position in frequency-dependent and temperaturedependent experiments.

Such a behaviour was previously identified by Monte-Carlo simulations of frequency-dependent internal friction [START_REF] Maugis | Giant Snoek peak in ferrite due to carbon-carbon strain interactions[END_REF]. For technical reasons, the Monte-Carlo simulations were restricted to high stress amplitudes (100 MPa). Such amplitudes are more in line with the hysteretic damping phenomenon than with internal friction. Conversely, the thermokinetic model allows for small stress amplitudes. The present work shows that the above-mentioned relaxation properties are kept with stress amplitudes as low as 1 MPa. Our model has also the advantage of being very fast running, which allows for an easier study of the low-frequency regime.

Our understanding of the non-linear e ects on Snoek relaxation is in line with Zener's theory [START_REF] Zener | Elasticity and anelasticity of metals[END_REF]. It is at variance, however, to some previous approaches. Based on a Ginzburg-Landau approximation, Nowick and Berry studied theoretically the energy-loss singularity in the vicinity of a second-order transition, which is of Curie-Weiss type [START_REF] Nowick | Anelastic relaxation in crystalline solids[END_REF][START_REF] Nowick | Anelasticity and stress-induced ordering of point defects in crystals[END_REF]. Our approach is closer to that of Dattagupta et al. [START_REF] Dattagupta | Strain ordering in BCC metals and the associated anelasticity[END_REF], who recognised that the Zener order-disorder transition is a first-order transition. Our mathematical treatment is somewhat simpler that Dattagupta's. It clarifies one point: giant relaxation is related to the spinodal temperature T * via the diverging term (T *T * ) *1 , whereas the sharp peak at low frequency is related to the transition temperature T 0 . In the Fe-C system, these temperatures can be separated by 15 K or more, depending on the carbon content.

Comparison between the thermo-kinetic model and the linear-response theory highlights the limits of the linear approximation. Actually, only a non-linear approach can capture the influence of the order-disorder transition on the lowfrequency response. In simple words, the linear model does not know the order-disorder transition, although it knows the spinodal point.

The non-Debye character of our computed Snoek peaks is a manifestation of the so-called "interaction" between the solute atoms [START_REF] Blanter | Internal Friction in Metallic Materials[END_REF]. Contrary to the works of Weller et al. [START_REF] Weller | Internal Friction Studies on Oxygen-Oxygen Interaction in Niobium. I. Experimental Results and Application of Previous Interpretations[END_REF], Haneczok et al. [START_REF] Haneczok | Internal Friction Studies on Oxygen-Oxygen Interaction in Niobium[END_REF][START_REF] Haneczok | Interaction of interstitial solute atoms in bcc metals[END_REF] and Blanter et al. [START_REF] Blanter | Solute interaction and internal friction spectra in solid solutions[END_REF][START_REF] Blanter | Strain-induced interaction of dissolved atoms and mechanical relaxation in solid solutions. A review[END_REF][START_REF] Blanter | Interstitial-interstitial interactions in bcc VB group metals: Ab initio calculations[END_REF], the short-range interactions are not at work in our approach. The long-range elastic interaction between the defect dipoles and the homogeneous strain is rather the origin of both the peak increase and the peak shift. The homogeneous strain is generated by both the applied stress and the carbon-induced strain. As a consequence of the carbon-induced strain, the carbon atoms are not independent: once a carbon atom has moved to a favoured site, it produces a strain that further favours energetically the sites of same type. The resulting autocatalytic e ect is all the more pronounced the closer to the order-disorder spinodal. Near the spinodal point, applying a small stress at the adequate frequency is enough to promote a large proportion of carbon atoms into the favoured sites. This collective behaviour of interstitials atoms creates the giant anelastic response. Theories that approximate the total relaxation strength to the sum of individual atomic contributions do not account for this collective behaviour. They can therefore not predict giant relaxation.

The carbon-strain interaction acts not only on the carbon atoms sitting in stable octahedral positions. The interaction also modifies the energetic path of migrating carbon atoms, especially at the tetrahedral unstable position. As a consequence, the activation energy of the relaxation time is composition-dependent. This further raises the relaxation time at high carbon content, in addition to the e ect of the spinodal.

We found that both characteristic temperatures T 0 and T * are linear functions of the carbon content. Such a composition dependence was identified experimentally by Haneczok et al. [START_REF] Haneczok | Internal Friction Studies of Carbon-Carbon Interaction in Iron[END_REF] on the so-called critical temperature in the Fe-C system. Besides, a non-linear increase in the peak height was evidenced when increasing solute oxygen in Ta [START_REF] Weller | Internal friction study on the existence of oxygen pairs in interstitial solid solution of tantalum with oxygen[END_REF]. The origin of this is probably not the same as in our case because the measured curvature of the Q *1 m -oxygen content curve is opposite to ours. In addition, the dipole tensor of the Ta-O system is probably too weak to produce a measurable e ect of the dipole-strain interaction in this system. The experimental confirmation of the non-linear dependence of Q *1 m on solute carbon in Fe-C is still awaited. The sharp peak predicted at the crossing of temperature T 0 during cooling is not reported in the literature. Two reasons can explain this: first, according to our calculations the peak only occurs at low-frequency oscillations (f Õ 10 *1 Hz), which are rarely used in temperature-dependent internal friction. Second, the calculated peak has a very narrow temperature range, and may be damped out if the cooling rate is too fast. For these reasons, the peak will be di cult to evidence, unless specific means are implemented.

Conclusion

Based on the elasticity theory of point defects and on the rate theory of thermally activated processes, we built a mathematical model of the kinetics of Snoek relaxation. Its application to frequency-dependent and temperature-dependent internal friction in the Fe-C system showed that the peak in energy loss grows non-linearly with carbon content. In the vicinity of the spinodal of the Zener disorder to order transition, the energy loss exhibits giant values. The peak position is shifted by the thermodynamic e ect of the transition and by the kinetic e ect of the composition-sensitive carbon migration path.

Figure 1 :

 1 Figure 1: Snoek relaxation profile from the thermo-

1 .Figure 2 :

 12 Figure 2: Snoek relaxation profiles for various compositions computed with the thermo-kinetic model (lines) compared to the simulations by kinetic Monte Carlo (circles) [21]. The temperature is T = 300 K.

Figure 3 :

 3 Figure 3: Top: stress-strain hysteresis cycles corresponding to the peak frequencies of Figure 2. Bottom: quasi-static stressstrain isotherms.

Figure 4 :

 4 Figure 4: Top: temperature-stress phase diagram computed at carbon content C = 1.37%. The spinodal point lies at = 0 and T * = 298 K. Bottom: stress-strain isotherms computed at C = 1.37%. Note that the stress axis is horizontal in both graphs.

Figure 5 :

 5 Figure 5: Snoek relaxation profiles computed with the kinetic model at various oscillation frequencies (lines) compared to the linear model (dashed lines), for C = 1.37%. The linear model fails in the low-frequency regime.

Figure 6 :

 6 Figure 6: Snoek relaxation profiles for various carbon contents computed with the linear model. The oscillation frequency is f = 1 Hz.

Figure 7 :Figure 8 :

 78 Figure 7: Peak energy loss and peak temperature of the Snoek profile as functions of carbon content, computed from the linear model. f = 1 Hz.

Aknowledgements

This work was supported by the Agence Nationale de la Recherche (contract C-TRAM ANR-18-CE92-0021).

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.