AN IMAGE OF INERTIA ARGUMENT FOR ABELIAN SURFACES AND FERMAT EQUATIONS OF SIGNATURE (13, 13, n) 1. Introduction

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1 AN IMAGE OF INERTIA ARGUMENT FOR ABELIAN SURFACES AND FERMAT EQUATIONS OF SIGNATURE (13, 13, n) NICOLAS BILLEREY, IMIN CHEN, LASSINA DEMBÉLÉ, LUIS DIEULEFAIT, AND NUNO FREITAS Abstract. Building on previous work, we show that, for any integer n 2, the equation x 13 + y 13 = 3z n has no non-trivial solutions. For this, we need to deal with the obstruction which arises from the fact that the 7-torsion of one of the Frey curves associated to this equation is a Galois submodule of the 7-torsion of the Jacobian of a certain genus 2 hyperelliptic curve C. We remove this obstruction by combining the modularity of the Jacobian of C with an image of inertia argument applied to that surface. 1. Introduction In [5], 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 due to a single Hilbert modular form g of weight 2 and level N = (2 3 13) over Q( 13) with coefficients in Q( 2). In this paper, we provide two extensions of the modular method to eliminate the form g; as a consequence, we obtain a complete resolution of the generalized Fermat equations in (1.1). Theorem 1.1. For all integers n 2, there are no integer solutions (a, b, c) to the equation such that abc 0 and gcd(a, b, c) = 1. x 13 + y 13 = 3z n The first method is to eliminate the form g by extending the 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 (see [5, Remark 7.4]) that the obstructing form g is congruent modulo 7 to the Hilbert newform associated to the Frey curve corresponding to the trivial solution (1, 1, 0). Establishing this congruence solves (1.1) for p = 7 given that all the Date: February 12, Mathematics Subject Classification. Primary 11D41; Secondary 11G10, 11F80. Key words and phrases. Fermat equations, abelian surfaces, modularity, Galois representations. We acknowledge the financial support of ANR-14-CE Gardio (N. B.), an NSERC Discovery Grant (I. C.), and the grant Proyecto RSME-FBBVA 2015 José Luis Rubio de Francia (N. F.). 1

2 remaining arguments in [5] carry over. However, 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. It is natural to wonder whether one could simply use Chabauty methods to solve (1.1) for p = 7 along the lines of [11]. 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 Q(ζ 13 ). However, since a is not a seventh power in F, such calculations would in fact require working over the extension F ( 7 a) which is of 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). Therefore, the results in this paper 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, but also that it can be the more effective method even when dealing with small exponents. The computations required to support the proof of our theorems were performed using Magma [6]. The program files are available at [4]. Acknowledgements. We would like to thank T. Dokchitser and C. Doris for making their algorithm available to us, and for helpful correspondence. We also thank Michael Stoll for conversations regarding applying the Chabauty method for the problem treated in this paper. 2. The abelian surface A g attached to the form g 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 (2.1) C y 2 = (32u + 36)x 6 + (24u + 40)x 5 + ( u 32)x 4 + ( 16u + 8)x 3 + (17u 28)x 2 + ( 6u + 16)x + 6u 16, and we denote its Jacobian by J. Lemma 2.1. 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 [6] to compute the odd part of the conductor of J, and the Dokchitser- Doris algorithm [16] 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 [27, 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 [27, Proposition 2] and [27, 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). 2

3 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 +ζ13 1 ) be the maximal totally real subfield of Q(ζ 13 ), the cyclotomic field of 13th roots of unity. 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 2.2. The surface J acquires good reduction over K w. Proof. Using Magma [6], 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 [5, 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 [28, Proposition ] or [32, Theorem B]). So, 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 2.3. Let C and J be given by (2.1). Then, we have the following: (1) End K (J) = Z[ 2], i.e. J is of GL 2 -type with real multiplication (RM) by Z[ 2]; (2) 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 [17, Theorem 17], there is an equation for the Humbert surface H 8 of discriminant 8, which parametrises principally polarized abelian surfaces with RM by Z[ 2]. It gives the surface H 8 as a double-cover of the weighted projective space P 2 r,s. The point (r, s 20u u ) = (, ) determines the isomorphism class of an abelian surface A whose Igusa-Clebsch invariants are given by I u = ; 81 I u = ; 6561 I u = ; I u = Let I 2, I 4, I 6 and I 10 be the Igusa-Clebsch invariants of the surface J, and α = 60u 48. Then, we have I 2i = α 2i I 2i for i = 1, 2, 3, 5. Thus the surfaces A and J are isomorphic over K. Since A has RM by Z[ 2], so does J. This gives (1). Alternatively, one could compute the endomorphism ring End K (J) using the algorithm in [10]. We now prove (2). 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)). 3

4 Let ρ J,3 be the mod 3 reduction of this representation. By construction, ρ J,3 is odd. Since, J is principally polarized, and of GL 2 -type, it follows from Ribet [29, Proposition 3.3] that, for q 3N, the characteristic polynomial at q is of the form charpoly(ρ J,3 (Frob q )) = x 2 a q x + Nq, where Tr(ρ J,3 (Frob q )) = a q Q( 2), and Nq is the norm of q. By reduction modulo 3, we see that the image of ρ J,3 is contained in {g GL 2 (F 9 ) det(g) 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 ρ J,3 (Frob q ) for the primes q above 17 and 53, we see that the projective image of ρ J,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 ρ J,3 is PSL 2 (F 9 ). In particular, we see that the image of ρ J,3 contains SL 2 (F 9 ), so ρ J,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 v1 )) = 2 ± 2, and Tr(ρ J,3 (Frob v2 )) = ± 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 ρ J,3 I5 is trivial, hence it has odd order. Since 3 and 5 have odd ramification indices in K, it follows that ρ J,3 satisfies the conditions of [19, Theorem 3.2 and Proposition 3.4]. Hence, it is modular. We use [24, Theorem 3.5.5] to conclude that ρ J,3, and hence J, is modular. By local-global compatibility ([9, Théorème (A)]) and Lemma 2.1, 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. 3. The image of inertia argument with a surface 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 [1], has been extensively applied and refined (see [5, 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 2.3 to describe (see Theorem 3.3) the inertial type of the non-rational form g at the prime (2); this together with the local information at (w) given by Lemma 2.2 allows for a proof of Theorem

5 Before proceeding we need some notation. Let E = E 1, 1 be the Frey curve attached to the trivial solution (1, 1, 0) in [5, Section 7.1]; it admits a minimal model given by E y 2 = x 3 ux 2 + (9u 25)x 17u Let also J be as above and p 7 be the prime of Q( 2) above 7 generated by Let K 2 /Q 2 be the unique unramified quadratic extension and K2 un 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 /K2 un where A obtains good reduction. By a result of Serre-Tate ([30, 2, Corollary 3]), we have M A = K2 un (A[p]) for any odd prime p. We recall that the curve E has potentially good reduction at 2. By Lemma 2.1, the same is true for J; so M E = K2 un(e[3]) and M J = K2 un(j[3]). Proposition 3.1. 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))) = (5, 5, 4). Hence υ 2 (j(e)) = 11 and E has potentially good reduction at (2). It follows from [8, pp. 675, Corollaire] that E has semistability defect e = 24, hence Gal(M E /K un 2 ) SL 2(F 3 ) by [25]. On the other hand, J also has potentially good reduction at (2) by Lemma 2.1, and as a byproduct, the Dokchitser-Doris algorithm [16] returns the totally ramified field K 2 (J[3]) with the defining polynomial h J = x 24 + (87α + 131)2 2 x 23 + (15α + 114)2 3 x 22 + (115α 206)2x 21 where α K 2 satisfies α 2 + α + 1 = 0. + ( 360α + 475)2x 20 + (115α 315)2x 19 + ( 115α + 76)2x 18 + ( 133α + 231)2 2 x 17 + (460α 17)2x 16 + (72α + 195)2 2 x 15 + (39α 385)2x 14 + (489α + 67)2x 13 + (455α + 171)2x 12 + ( 232α 251)2 2 x 11 + (187α 103)2 2 x 10 + ( 7α 508)2x 9 + ( 31α + 18)2 4 x 8 + (29α + 47)2 3 x 7 + (465α + 332)2x 6 + ( 203α 14)2 2 x 5 + (14α + 107)2 2 x 4 + ( 46α + 7)2 3 x 3 + (142α + 249)2 2 x 2 + (35α 107)2 3 x + ( 423α + 188)2, We check that E has good reduction over K 2 (J[3]), so M E M J = K2 un (J[3]) by minimality. Since [M E K2 un] = 24 = [M J K2 un ] 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 3.2. 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 g be a Hilbert modular form over K of level N and field of coefficients Q g. Assume that q N and that g has a supercuspidal exceptional type at q. Then, for all primes p coprime to 6N and all primes P p in Q g, we have ρ g,p (I q ) SL 2 (F 3 ). 5

6 Proof. Let π g be the automorphic representation attached to g, and π g,q the local component at q. Also, let σ g,q W q GL 2 (C) be the Weil representation attached to π g,q by the local Langlands correspondence (see [26]). Since π g,q is a supercuspidal exceptional representation, then σ g,q is an exceptional representation, which means that the projective image of σ g,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 g. Let D q I q be a decomposition group at q in G K. By local-global compatibility ([9, Théorème (A)]), the projective image of ρ g,p Dq in PGL 2 (Q p ) is either A 4 or S 4, and ρ g,p Iq acts irreducibly [7, 42.1] (in loc. cit., supercuspidal exceptional representations are called primitive representations). Since p 6, the image of ρ g,p Dq in PGL 2 (F p ) is also A 4 or S 4, and ρ g,p Iq acts irreducibly. A careful analysis of the proof of [14, Proposition 2.4] shows that it carries over to any finite local extension of Q 2. In particular, this implies that the projective image of ρ g,p Iq is equal to A 4. Therefore, the image of ρ g,p Iq SL 2 (F p ) is isomorphic to either A 4 or SL 2 (F 3 ); the result now follows since there is only one element of order 2 in SL 2 (F p ) (for p > 2), hence no subgroup of SL 2 (F p ) is isomorphic to A 4. Theorem 3.3. Let P 7 in F = Q( 2) be a prime. Then, we have and, moreover, M J = K un 2 (J[7]) = K un 2 (J[P]) ρ g,p I2 ρ E,7 I2. 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 construction, we have J[7] = J[P] J[P ]. This means that the field M J = K2 un (J[7]) is the compositum of Kun 2 (J[P]) and Kun 2 (J[P ]), the fields cut out by ρ J,P I2 and ρ J,P I2 respectively. We will now show that these three fields have the same degree and the first statement follows. By Theorem 2.3, we have ρ J,P ρ g,p and ρ J,P ρ g,p. By Proposition 3.1 and the discussion preceding it, we only need to show that the fields cut out by ρ g,p I2 and ρ g,p I2 have degree # SL 2 (F 3 ) = 24. Note that if ρ g,p I2 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 I2 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 I2 would have projective dihedral image; since the field cut out by ρ g,p I2 is a Galois subextension of K2 un(j[7])/kun 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 be have a supercuspidal exceptional type at 2. By Lemma 3.2, we obtain that the fields cut out by ρ g,p I2 have degree # SL 2 (F 3 ) = 24 as required. We now prove the last statement. From the first part of the theorem, Theorem 2.3 and Proposition 3.1, we have that M J is the field cut out by both ρ g,p I2 and ρ E,7 I2. Thus, ρ g,p I2, 6

7 and ρ E,7 I2 have the same kernel, and image SL 2 (F 3 ) SL 2 (F 7 ) GL 2 (F 7 ). Therefore, it follows from [21, Lemma 2] that they are isomorphic representations. Finally, we now prove our Diophantine result using the information on the inertial types. First proof of Theorem 1.1. Clearly, it suffices to consider the case when n = p is a prime. The case p /= 7 is [5, Theorem 2]. To establish the remaining case p = 7, we note that the rest of the arguments will carry over to yield [5, Theorem 2] provided that we can show that [5, Theorem 7] is still true in this case. Let E a,b be the Frey curve defined in [5, Section 7.1]. The proof of [5, Theorem 7] uses [5, Proposition 9] which asserts that ρ Ea,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 Ea,b,7 ρ g,p where P 7 in Q( 2). By [5, Remark 7.4] we have P = p 7. Suppose p = 7, and Z is one of the three curves above, the arguments of [5, Theorem 7] still hold. For instance, if Z = E 1, 1 and 4 a + b, then it is shown that (3.1) ρ Ea,b,p I2 / ρ Z,p I2. By Theorem 2.3, we have ρ g,p7 ρ J,p7. Therefore, to complete the proof of [5, Theorem 7], we only have to eliminate the possibility that ρ Ea,b,7 ρ g,p7. The proof of [5, Theorem 7 (A)] remains valid by simply replacing Z/K with J/K w and using Lemma 2.2. To get Part (B), we note that ρ g,p7 I2 ρ E1, 1,7 I2 by Theorem 3.3, and that ρ E1, 1,7 I2 / ρ Ea,b,7 I2 by (3.1), for 4 a + b. Remark 3.4. 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. Hence, the methods in [13] and [17] 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. One can thus expect the methods in this paper 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 surface with real multiplication and reasonable height. 4. Proof of Theorem 1.1 using congruences In this section, we outline a proof of Theorem 1.1 which uses the mod 7 congruence mentioned in the introduction. This proof can be seen as one of global nature given that it uses the global Galois representations in the 7-torsion points of the surface J, and the Frey curve E. Let E = E 1, 1, J and p 7 be as in the previous section; in particular, we have F p7 = F 7. Let M = S 2 (N) new be the new subspace of Hilbert cusp forms of parallel weight 2, trivial character, and level N = (2 3 w 2 ). Theorem 2.3 shows that J is modular and corresponds to the Hilbert newform g which lies in M. The elliptic curve E is modular by [22] and corresponds to a newform f M. Thus, ρ J,p7 ρ g,p7 and ρ E,7 ρ f,7. 7

8 Proposition 4.1. Consider the residual representations ρ g,p7 Gal(Q/K) GL 2 (F 7 ) and ρ f,7 Gal(Q/K) GL 2 (F 7 ). Then, we have ρ g,p7 ρ f,7. Proof. Let M = S 2 (N, F 7 ) new. Let T T End(M) be the Hecke algebras generated by the Hecke operators T q for prime ideals q 7N of norm up to 200, and all the Hecke operators T q for prime ideals q 7N, respectively. Let S be the socle of M considered as a T -module, i.e. the sum of all simple T -submodules of M, which is semi-simple. Let ḡ and f be the reductions of the eigenforms g and f modulo p 7 and 7, respectively. Both ḡ and f are eigenforms, lie in M, and generate two simple T -modules Wḡ and W f which have dimension one over F 7, respectively. Hence, Wḡ and W f are contained in S. They are isomorphic as T -modules as for all prime ideals q 7N of norm up to 200, we have that a q (g) (mod p 7 ) = a q (f) (mod 7) as elements in F 7. Using Magma [6], 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, Wḡ = W f inside S M. Since Wḡ = W f inside M and both are T-modules as well, we obtain that a q (g) (mod p 7 ) = a q (f) (mod 7) as elements of F 7, for all prime ideals q 7N. Remark 4.2. This method of establishing ρ g,p7 ρ f,7 without the use of a Sturm bound should work for more general g and f, provided one can establish that the simple constiuents of the socle S have multiplicity one. Second proof of Theorem 1.1. The bulk of the argument in Section 4, including the identity (3.1), still applies. Therefore, we only need to show that the isomorphism ρ Ea,b,7 ρ g,p7 is not possible. However, by Theorem 2.3 and Proposition 4.1, we have ρ Ea,b,7 ρ g,p7 ρ J,p7 ρ E1, 1,7. The arguments that eliminated E 1, 1 apply to deal with g, completing the proof. References [1] M. A. Bennett and C. M. Skinner, Ternary Diophantine equations via Galois representations and modular forms, Canad. J. Math. 56 (2004), no. 1, [2] N. Billerey, Équations de Fermat de type (5, 5, p), Bull. Austral. Math. Soc. 76 (2007), no. 2, [3] N. Billerey and L. Dieulefait, Solving Fermat-type equations x 5 +y 5 = dz p, Mathematics of Computation 79 (2010), no. 286, [4] N. Billerey, I. Chen, L. Dembélé, L. Dieulefait, and N. Freitas, Supporting Magma program files for this paper, 1 [5] N. Billerey, I. Chen. L. Dieulefait, and N. Freitas, A multi-frey approach to Fermat equations of signature (r, r, p), Transactions of AMS (to appear). 1, 1, 2, 3, 3, 3 [6] W. Bosma, J. Cannon, and C. Playoust. The Magma algebra system. I. The user language. J. Symbolic Comput., 24(3-4): , Computational algebra and number theory (London, 1993). 1, 2, 2, 4 [7] C. J. Bushnell, G. Henniart, The local Langlands conjecture for GL(2). Grundlehren der Mathematischen Wissenschaften 335, Springer-Verlag, Berlin, xii+347 pp. 3 8

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10 Université Clermont Auvergne, CNRS, LMBP, F Clermont-Ferrand, France. address: Department of Mathematics, Simon Fraser University, Burnaby, BC V5A 1S6, Canada address: Department of Mathematics, King s College London, Strand, WC2R 2LS, London, UK address: lassina.dembele@gmail.com Departament d Algebra i Geometria, Universitat de Barcelona, G.V. de les Corts Catalanes 585, Barcelona, Spain address: ldieulefait@ub.edu Mathematics Institute, University of Warwick, Coventry CV4 7AL, United Kingdom address: nunobfreitas@gmail.com 10