On Fermat’s equation over some quadratic imaginary number fields
 George C. Ţurcaş^{1}Email authorView ORCID ID profile
Received: 1 November 2017
Accepted: 25 April 2018
Published: 4 May 2018
Abstract
Assuming a deep but standard conjecture in the Langlands programme, we prove Fermat’s Last Theorem over \({\mathbb {Q}}(i).\) Under the same assumption, we also prove that, for all prime exponents \(p \ge 5,\) Fermat’s equation \(a^p+b^p+c^p=0\) does not have nontrivial solutions over \({\mathbb {Q}}(\sqrt{2})\) and \({\mathbb {Q}}(\sqrt{7}).\)
Keywords
Mathematics Subject Classification
1 Introduction
Wiles’ remarkable proof of Fermat’s Last Theorem inspired mathematicians to attack the Fermat equation over number fields via elliptic curves and modularity. Successful attempts over totally real quadratic fields had been carried out by Jarvis and Meekin [20], Freitas and Siksek [11, 12] and they rely on progress in modularity over totally real fields due to work of Barnett–Lamb, Breuil, Diamond, Gee, Geraghty, Kisin, Skinner, Taylor, Wiles and others. In particular, modularity of elliptic curves over real quadratic fields was proved by Freitas et al. [13].
The present work follows the skeleton of their article. Specialising to the fields \({\mathbb {Q}}(i),\, {\mathbb {Q}}(\sqrt{2})\) and \(\mathbb Q(\sqrt{7}),\) we were able to make their result effective and to obtain optimal bounds for the constant \(B_K.\) Combining this with previous work on the Fermat equation over number fields, was sufficient to derive Theorem 1.1 below. Şengün and Siksek [27, Conjecture 4.1] is known to hold for basechange newforms via the theory of base change functoriality, therefore we obtain a result that is dependant only on Serre’s modularity conjecture (see Conjecture 2.1).
 (1)
First, we had to obtain a small enough constant \(B_K\) such that for \(p> B_K\) the mod p Galois representation that comes from a Frey curve is absolutely irreducible. This is treated asymptotically in [27], but we had to rely on upper bounds for possible prime torsion that can be achieved by elliptic curves defined over number fields of small degree and class field theory computations to make the bound \(B_K\) explicit. We should point out that the class field theory computations are different in an essential way from the ones that had been done in [11, 24]. The latter rely on the presence of nontrivial units in \({\mathcal {O}}_K,\) which are not available in our setting.
 (2)
To keep this introduction short, we defer the precise definitions of eigenforms and mod p eigenforms to Sect. 2. Serre’s modularity conjecture relates certain representations \(G_K \rightarrow {{\mathrm{GL_2}}}(\overline{{\mathbb {F}}}_p)\) to a mod p eigenform of weight 2 over K. In the classical situation Serre’s modularity conjecture, now a theorem due to Khare and Wintenberger, relates some representations \(G_{{\mathbb {Q}}} \rightarrow {{\mathrm{GL_2}}}(\overline{{\mathbb {F}}}_p)\) to a complex eigenform of weight 2 over \({\mathbb {Q}}.\) The expression “mod p” does not appear there, because it turns out that all mod p eigenforms over \({\mathbb {Q}}\) are just reductions of complex eigenforms. This is not the case if K is imaginary quadratic, but Şengün and Siksek [27] managed to circumvent this difficulty by assuming that the prime p is large enough. Unfortunately this step makes their result ineffective. For our choices of K, we made this step effective by computing the torsion in certain cohomology groups using an algorithm of Şengün. This allowed us to lift the mod p eigenforms to complex ones.
 (3)
A complex weight 2 newform over K with rational eigenvalues corresponds conjecturally to an elliptic curve or a fake elliptic curve defined over K (see [27]). The theory of base change functoriality shows that this conjecture holds for the newforms that are basechange lifts of classical newforms. By explicit computation we find that all the newforms that we need to deal with are basechange lifts.
From now on, we restrict ourselves to \(K \in \{ {\mathbb {Q}}(i),\, {\mathbb {Q}}(\sqrt{2}),\, {\mathbb {Q}}(\sqrt{7}) \}\) and we justify this restriction to the reader.
One reason is that we had to carry out some explicit computations in the cohomology groups of locally symmetric spaces (see Sect. 2) that were much simpler when the field had class number 1.
To carry out our argument and prove that there are no solutions to Fermat’s equation, we had to prove that Galois representations associated to Frey curves are absolutely irreducible. In doing so, we made use of the fact that K has primes of residual degree 1 above 2 so we end up with the three number fields above. On top of that, one should be aware of the existence of solutions \(1^p + (\omega )^p + (\omega ^2)^p =0,\) where \(\omega \) is a primitive third root of unity and p is an odd prime. These are defined over \({\mathbb {Q}}(\sqrt{3}),\) making a proof of Fermat’s Last Theorem over \({\mathbb {Q}}(\sqrt{3})\) hopeless.
Theorem 1.1
Work is in progress to prove similar statements to Theorem 1.1 for the other imaginary quadratic fields of class number one, but in those cases at the moment we are just able to prove that there are no solutions to (1.1) such that Norm(abc) is even. We also hope to extend the methods to prove similar statements for other quadratic imaginary fields of nontrivial class number in the spirit of [11].
Corollary 1.2
The reader might now legitimately ask themselves why we were only able to derive this corollary for the case \(K= {\mathbb {Q}}(i)\) and we will try to explain this now. In fact, our techniques prove the result claimed in Theorem 1.1 for \(p \ge 19\) when \(K \in \{ {\mathbb {Q}}(i),\, {\mathbb {Q}}(\sqrt{2}) \}\) and \(p \ge 17\) when \(K={\mathbb {Q}}(\sqrt{7})\) and we rely on previous work on Fermat’s Last Theorem over number fields to conclude the same result for small values of p. To be precise, a result of Gross and Rohrlich [15, Theorem 5.1] implies that there are no solutions to (1.1) for \(p=5,\,7,\,11\) and our three choices of K. The case \(p=13\) is covered completely by work of [33].
Hao and Parry [18] also worked on Fermat’s equation over quadratic fields and one of their results [18, Theorem 4] can be used to complete the remaining cases \(p=17\) and \(K \in \{ {\mathbb {Q}}(i),\, {\mathbb {Q}}(\sqrt{2}) \}\) of our Theorem 1.1. All these works do not depend on the aforementioned conjecture of Serre.
It is worth pointing out that if one knows that none of the equations \(a^p +b^p + c^p =0\) for \(p \ge 3\) prime and \(a^4+b^4=c^4\) has a solution in \((a,\,b,\,c) \in (K{\setminus } \{0\})^3,\) then one can deduce results analogous to Corollary 1.2 for K. In fact that is how the corollary is proved.
The equation \(a^3+b^3+c^3=0\) describes a curve of genus 1 and can be seen as an elliptic curve over K after a choice of basepoint. This elliptic curve has Mordell–Weil group isomorphic to \({\mathbb {Z}}/ 3 {\mathbb {Z}}\) when \(K \in \{ {\mathbb {Q}}(i),\, {\mathbb {Q}}(\sqrt{7}) \}.\) The torsion points correspond to the trivial solutions (i.e., one of \(a,\,b,\,c\) is 0) to Fermat’s equation under reparameterization. The situation is a little bit different when \(K= {\mathbb {Q}}(\sqrt{2}),\) because the elliptic curve in play has rank 1 and the nontorsion points on this curve correspond to infinitely many solutions \((a,\,b,\,c) \in {\mathbb {Q}}(\sqrt{2})^{*}\) to \(a^3+b^3+c^3=0.\) All the computations here have been carried with the elliptic curve packages available in Magma [3]. For the equation \(a^4+b^4=c^4,\) we are going to refer to the work of Aigner [1] who proved that the only nontrivial solutions to this equation over quadratic imaginary fields are defined over \({\mathbb {Q}}(\sqrt{7}).\) Note that \((1+ \sqrt{7})^4+ (1\sqrt{7})^4 = 2^4\) is one of them. One can see now how the above discussion together with Theorem 1.1 imply the corollary and also why such a corollary cannot be achieved for any of the other two fields that are covered in the theorem.
2 Eigenforms for \({{\mathrm{GL_2}}}\) over K and Serre’s modularity conjecture
The exposition in this section follows closely along the lines of [21, 27]. A celebrated theorem of Khare and Wintenberger connects certain 2dimensional continuous representations of \(G_{{\mathbb {Q}}}\) into \({{\mathrm{GL_2}}}(\overline{{\mathbb {F}}}_p)\) with classical modular forms of the hyperbolic plane \({\mathcal {H}}_2.\) In this section we are going to discuss a conjecture that aims to generalise the theorem of Khare and Wintenberger in a way that is going to be soon clarified.
Let p be a prime that does not ramify in K and is coprime to \({\mathfrak {N}}.\) The cohomology groups \(H^{i}(Y_0({\mathfrak {N}}),\, \overline{{\mathbb {F}}}_p)\) come equipped with commutative Hecke algebras \({\mathbb {T}}_{{\mathbb {F}}_p}^i({\mathfrak {N}}),\) generated by Hecke operators \(T_{\mathfrak \pi }\) associated to the prime ideals \((\pi )\) of \({\mathcal {O}}_K\) away from \(p {\mathfrak {N}}.\)
By a mod p eigenform \(\Phi \) (over K) of level \({\mathfrak {N}}\) and degree i we refer to a ring homomorphism \(\Phi : {\mathbb {T}}_{{\mathbb {F}}_p}^i({\mathfrak {N}}) \rightarrow \overline{{\mathbb {F}}}_p.\) It is known that the values of a mod p eigenform generate a finite extension \({\mathbb {F}}\) of \({\mathbb {F}}_p.\) Two such mod p eigenforms of levels \({\mathfrak {N}},\, {\mathfrak {M}}\) are said to be equivalent if their values agree on the Hecke operators associated to prime ideals away from \(p {\mathfrak {N}} {\mathfrak {M}}.\)
Consider the complex cohomology groups \(H^{i}(Y_0({\mathfrak {N}}),\, {\mathbb {C}}).\) For all prime ideals \((\pi ) \subset {\mathcal {O}}_K,\) we can construct linear endomorphisms \(T_{\pi }\) of \(H^i(Y_0({\mathfrak {N}}),\, {\mathbb {C}}),\) called Hecke operators. They form a commutative Hecke algebra denoted \({\mathbb {T}}^i_{{\mathbb {C}}}({\mathfrak {N}}).\) By a complex eigenform f over K of degree i and level \({\mathfrak {N}}\) we mean a ring homomorphism \(f:{\mathbb {T}}^i_{{\mathbb {C}}}({\mathfrak {N}}) \rightarrow {\mathbb {C}}.\) The values of f are algebraic integers and they generate a number field \({\mathbb {Q}}_f.\) Two eigenforms \(f,\,g\) are equivalent if their values agree on the Hecke operators. One should notice that since two equivalent eigenforms can be of different levels, the Hecke operators on which they agree might live in different Hecke algebras. A complex eigenform, say of level \({\mathfrak {N}},\) is called new if it is not equivalent to one whose level is a proper divisor of \({\mathfrak {N}}.\) A complex eigenform f is called trivial if \(f(T_{\pi })=N_{K/{\mathbb {Q}}}(\pi ) +1\) for all prime ideals \((\pi )\) of \({\mathcal {O}}_K.\) In the setting of \({{\mathrm{GL_2}}},\) nontriviality amounts to cuspidality.
We have to point out that the virtual cohomological dimension of \(Y_0({\mathfrak {N}})\) is 2 and for \(p \ge 5,\) the cuspidal parts of \(H^1(Y_0({\mathfrak {N}}),\, \overline{{\mathbb {F}}}_p)\) and \(H^2(Y_0({\mathfrak {N}}),\,\overline{{\mathbb {F}}}_p)\) are isomorphic as Hecke modules and the same holds for the cuspidal parts of \(H^1(Y_0({\mathfrak {N}}),\, {\mathbb {C}}))\) and \(H^2(Y_0({\mathfrak {N}}),\, {\mathbb {C}}).\) The latter follows from a duality result of Ash and Stevens [2, Lemma 1.4.3]. Therefore, in our setting, any mod p or complex eigenform is equivalent to one of the same level and degree 1. This particular case of a more general conjecture of Calegari and Emerton (see [5]) holds in the present situation due to the low dimension.
Since everything happens in degrees 1, 2 and the nontrivial parts of the corresponding Hecke algebras are isomorphic, we are going to forget about the upper script i and write \(\mathbb T_{{\mathbb {F}}_p}({\mathfrak {N}})\) respectively \({\mathbb {T}}_{\mathbb C}({\mathfrak {N}}),\) sometimes referring to the nontrivial part of the degree one Hecke algebra and sometimes referring to the one of degree two. The eigenforms in these algebras are sometimes called mod p Bianchi modular forms, respectively Bianchi modular forms in the literature.
It is discussed in [8] how, via the Eichler–Shimura–Harder isomorphism, Bianchi modular forms can be analytically interpreted as vector valued realanalytic functions on the hyperbolic 3space. We are going to be using this point of view in our examination of the Mellin transform of a Bianchi modular form in Sect. 5.
Given a complex eigenform f and a prime p that is coprime to the level of f, one can fix a prime ideal \({\mathfrak {p}}\) of \(\mathbb Q_f\) that lies above p and obtain a mod p eigenform \(\Phi _f\) (of the same level and degree) by setting \(\Phi _f(T_{\pi })= f(T_{\pi }) \pmod {{\mathfrak {p}}} \) for all prime ideals \((\pi )\) coprime to \(p{\mathfrak {N}}.\) We say that a mod p eigenform \(\Phi \) lifts to a complex one if there is a complex eigenform f with the same level and degree such that \(\Phi =\Phi _f.\)
In the above, the presence of ptorsion in \(H^2(Y_0({\mathfrak {N}}),\, {\mathbb {Z}})\) is the obstruction to surjectivity in the reduction map. The existence of an eigenform (complex or mod p) is equivalent to the existence of a class in the corresponding cohomology group that is a simultaneous eigenvector for the Hecke operators such that its eigenvalues match the values of the eigenform. With this interpretation, we can deduce that every mod p eigenform of degree 1 lifts to a complex one when \(H^{2}(Y_0({\mathfrak {N}}),\, {\mathbb {Z}})\) has no ptorsion.
We will be using a special case of Serre’s modularity conjecture over number fields. In his landmark paper [29], Serre conjectured that all absolutely irreducible, odd mod p Galois representation of \(G_{{\mathbb {Q}}}={{\mathrm{Gal}}}(\overline{{\mathbb {Q}}}/ \mathbb Q)\) arise from a cuspidal eigenform f. Here f is a classical cuspidal modular form. In the same article, Serre gave a recipe for the level N and the weight k of the sought after eigenform. As previously mentioned, this conjecture was proved by Khare and Wintenberger [23].
We are going to state a conjecture which concerns mod p representations of \(G_K,\) the absolute Galois group of the quadratic imaginary K.
Conjecture 2.1
Remark 2.2
When stated for more general number fields, this conjecture restricts to odd representations. A representation is odd if the determinant of every complex conjugation is \(1,\) but since our K is totally complex \(G_K\) does not contain any complex conjugations and we will regard every mod p representation of \(G_K\) automatically as odd.
Although it is conjecturally easy to predict the level \({\mathfrak {N}}\) of such an eigenform, doing the same thing for the weight can be very difficult. A quite involved general weight recipe for \({{\mathrm{GL_2}}}\) over number fields was given by Gee et al. [14]. We will just mention that this recipe depends on the restriction \(\overline{\rho }_{I_{{\mathfrak {p}}}}\) to the inertia subgroups for the primes \({\mathfrak {p}} \subset {\mathcal {O}}_K\) above p. We only considered very special representations \(\overline{\rho }\) (that are finite flat at \({\mathfrak {p}}p),\) for which Serre’s original weight recipe applies and predicts the trivial weight [29]. This is why we end up with classes in \(H^1(Y_0({\mathfrak {N}}),\, \overline{{\mathbb {F}}}_p),\) the trivial weight meaning that we get \(\overline{{\mathbb {F}}}_p\) as coefficient module.
3 Fermat equation with exponent p and the Frey curve
 p :

a rational prime number;
 K :

is one of \({\mathbb {Q}}(i),\) \({\mathbb {Q}}(\sqrt{2})\) or \(\mathbb Q(\sqrt{7});\)
 \({\mathcal {O}}_K\) :

the ring of integers of K;
 S :

set of prime ideals of \({\mathcal {O}}_K\) that lie above the prime 2.
From [12, Lemma 3.7] it follows that E has potentially multiplicative reduction at \({\mathfrak {a}} \in S\) and that \(p \mid \# \overline{\rho }(I_{{\mathfrak {a}}}),\) where \(I_{\mathfrak a}\) denotes the inertia subgroup of \(G_K\) at \({\mathfrak {a}}.\) This information is crucial for everything that follows.
The following is a particular case of [27, Lemma 5.4]. Since it is central to our discussion, we include a proof here.
Lemma 3.1
The Frey curve E is semistable away from S. The determinant of \(\overline{\rho }\) is the mod p cyclotomic character. Its Serre conductor \({\mathcal {N}}\) is supported on S and belongs to a finite set that depends only on the field K. The representation \(\overline{\rho }\) is finite flat at every prime \({\mathfrak {p}}\) of K that lies above p.
Proof
Let \({\mathfrak {q}} \not \in S\) be a prime of \({\mathcal {O}}_K.\) By wellknown results [30, Sect. VII], one deduces that the model (3.2) for E is minimal and the curve is semistable at \({\mathfrak {q}}.\)
The statement concerning the determinant is a well known consequence of the theory of the Weil pairing. \(\square \)
We denote by \(N_E\) the conductor of E and apply [11, Lemma 4.4] to this elliptic curve. Despite the fact that the results of [11] are stated for real quadratic fields, the arguments of this specific lemma are purely local and they continue to apply in our setting. We describe this process in details for \(K={\mathbb {Q}}(i)\) remarking that the computation is analogous for the other two number fields. Following the notation in [11], one can compute values of \(\lambda _i \in {\mathcal {O}}_K {\setminus } {\mathfrak {b}},\) which are some cokernel representatives of the map \(\Phi :{\mathcal {O}}_K^{*} \rightarrow \mathcal (O_K/ {\mathfrak {b}})^{*}/ (({\mathcal {O}}_K / {\mathfrak {b}})^{*})^2.\)
When \(K={\mathbb {Q}}(i),\) let \(2 {\mathcal {O}}_K ={\mathfrak {a}}^2.\) In the notation of [11], this gives \({\mathfrak {b}}= \mathfrak a^5,\) an ideal of norm 32. The image of \(\Phi :{\mathcal {O}}_K^{*} \rightarrow \mathcal (O_K/ {\mathfrak {b}})^{*}/ (({\mathcal {O}}_K / \mathfrak b)^{*})^2\) has order 2 and the codomain \((O_K/ \mathfrak b)^{*}/ (({\mathcal {O}}_K / {\mathfrak {b}})^{*})^2\) is isomorphic to \(({\mathbb {Z}}/ 2 {\mathbb {Z}} )^3.\) One can take as representatives of the cokernel of \(\Phi \) the elements \(\lambda _1 = 1,\) \(\lambda _2 =2+i,\) \(\lambda _3 = 3\) and \(\lambda _4=2+i\) in \({\mathcal {O}}_K.\) This gives \(\max \nolimits _{1 \le i \le 4} v_{{\mathfrak {a}}}(\Delta (K_{\mathfrak a}(\sqrt{\lambda _i})/K_{{\mathfrak {a}}}))=4.\) Even though one could do all the above computations using pen and paper only, the author made use of basic features of Magma at some steps.
Exponent conductor of E at \(\mathfrak a\)
Number field K  Factorization of \(2{\mathcal {O}}_K\)  Valuation of \(N_E\) 

\(\mathbb Q(i)\)  \(2{\mathcal {O}}_K = {\mathfrak {a}}^2\)  \(v_{\mathfrak a}(N_E) = 8\) 
\({\mathbb {Q}}(\sqrt{2})\)  \(2{\mathcal {O}}_K = {\mathfrak {a}}^2\)  \(v_{{\mathfrak {a}}}(N_E) = 8\) 
\({\mathbb {Q}}(\sqrt{7})\)  \(2{\mathcal {O}}_K = {\mathfrak {a}}_1 \cdot {\mathfrak {a}}_2, \, {\mathfrak {a}}_1 \ne {\mathfrak {a}}_2\)  \(v_{\mathfrak a_1}(N_E)=4 \text { and } v_{{\mathfrak {a}}_2}(N_E)=1\) 
\(v_{{\mathfrak {a}}_1}(N_E)=1 \text { and } v_{{\mathfrak {a}}_2}(N_E)=4\) 
4 Local computations and irreducibility of \(\overline{\rho }\)
For applying Conjecture 2.1 to the Galois representation \(\overline{\rho }= \overline{\rho }_{E,p},\) we have to prove that it is absolutely irreducible. We first prove irreducibility and the stronger condition will follow.
The conductor exponent of an elliptic curve is defined (see [31, Chap. IV]) as the sum between a tame and a wild part. Similarly, the conductor exponent of a Galois representation can be written as the sum between a tame and a wild part. In the following lemma, we think of a mod p character as a 1 dimensional representation over \({\mathbb {F}}_p\) and we define the wild part of its conductor exponent accordingly.
Lemma 4.1
Proof
The first equality follows from the fact that \(\theta \theta ' = \chi _{p},\) therefore when restricted to the absolute inertia group \(I_{{\mathfrak {q}}},\) the characters are inverses of each other.
In what follows, we are going to think of \(\theta \) and \(\theta '\) as one dimensional representations from \(G_K\) targeting the one dimensional subspaces of E[p] on which we see their actions.
Denote by \(K_{{\mathfrak {q}}}\) the completion of the number field K at the prime \({\mathfrak {q}}\) and by \(L=K_{{\mathfrak {q}}}(E[p])\) the extension of \(K_{{\mathfrak {q}}}\) that we get by adjoining the coordinates of the ptorsion of E.
Theorem 4.2
Let \(p \ge 19\) if \(K \in \{{\mathbb {Q}}(i) ,\, {\mathbb {Q}}(\sqrt{2}) \}\) and \(p \ge 17\) if \(K = {\mathbb {Q}}(\sqrt{7}).\) Then \(\overline{\rho }\) is irreducible.
Proof
The tame part of the exponent of a prime in the conductor of a character is at most 1 and must be equal 1 if the wild part is nonzero. On the other hand, the corresponding quantity in the conductor of an elliptic curve is at most 2 and must be equal to 2 if its wild part is nonzero. For the precise definitions, see [31, Chap. IV].
(i) Suppose p is coprime to \({\mathcal {N}}_{\theta }\) or \(\mathcal N_{\theta '}.\) Since the conductor of an elliptic curve is isogeny invariant, by eventually replacing E with the pisogenous curve \(E/\langle \theta \rangle \) we can assume that p is coprime to \(N_{\theta }\) and hence deduce that \(\theta \) is unramified away from the primes in S.
In turn, for \({\mathbb {Q}}(i)\) and \({\mathbb {Q}}(\sqrt{2}),\) \(\theta \) is a character of that corresponding ray class group, therefore the order of \(\theta \) divides the exponent of the group. If \(\theta \) has order 1 then E has a point of order p over K and we get a contradiction exactly as before. Similarly, if the order of \(\theta \) is 2 then E has a ptorsion point defined over a quadratic extension L of K. But \([L:{\mathbb {Q}}] = [L:K] \cdot [K:\mathbb Q]=4\) and the possible prime torsion of elliptic curves over quartic fields determined by Derickx et al. [10] would imply that \(p \le 17,\) a contradiction.
If \(\theta \) has order 4, then let L be the quadratic extension of K that is cut by the character \(\theta ^2.\) The restriction \(\phi =\theta _{G_L}\) is a quadratic character of \(G_L\) and therefore the twist by \(\phi \) of E, regarded as an elliptic curve over L, is an elliptic curve with a ptorsion point defined over L. The field L is of total degree 4 over \({\mathbb {Q}}\) and hence we get a contradiction as in the other case.
(ii) Suppose now that p is not coprime with \({\mathcal {N}}_{\theta }\) nor with \({\mathcal {N}}_{\theta '}.\) Since \(p \ge 19,\) we know that p is not ramified in K. If we suppose that p is inert in K, then from [24, Lemme 1] it follows that p divides only one of the conductors, a contradiction with the hypothesis.
The only case that has to be considered now is when p splits in K. Let \({\mathfrak {p}}_1,\, {\mathfrak {p}}_2\) be the two ideals of \({\mathcal {O}}_K\) such that \(p {\mathcal {O}}_K = {\mathfrak {p}}_1 \mathfrak p_2.\) We can suppose that \({\mathfrak {p}}_1  {\mathcal {N}}_{\theta },\) \({\mathfrak {p}}_1 \not \mid {\mathcal {N}}_{\theta '}\) and \({\mathfrak {p}}_2  {\mathcal {N}}_{\theta '},\) \({\mathfrak {p}}_2 \not \mid {\mathcal {N}}_{\theta }.\) The primes \({\mathfrak {p}}_1,\) \({\mathfrak {p}}_2\) are unramified so it follows from [28, Proposition 12] that E has good ordinary or multiplicative reduction at these primes and \(\theta _{I_{{\mathfrak {p}}_1}} = \chi _p_{I_{{\mathfrak {p}}_1}}\) and \(\theta '_{I_{{\mathfrak {p}}_2}} = \chi _p_{I_{{\mathfrak {p}}_2}}.\)
The character \(\theta ^2\) is unramified everywhere except \(\mathfrak p_1,\) because all the bad places for E are of potentially multiplicative reduction. We also know that \(\theta ^2_{ I_{{\mathfrak {p}}_1}} = \chi ^2_p_{I_{{\mathfrak {p}}_1}}.\)
We have used above the following class field theory lemma. We do not include a proof here since that would be mutatis mutandis, just a specialisation of [9, Proposition 2.4].
Lemma 4.3
Remark 4.4
For every place v, the map \(\iota _{v}\) is the inclusion of K into the completion \(K_v.\) Abusing notation we write \(i_{{\mathfrak {q}}}\) and \(K_{{\mathfrak {q}}}\) for the inclusion, respectively the completion of K, with respect to the corresponding prime ideal \({\mathfrak {q}}.\)
We obtain the following corollary, which implies absolute irreducibility.
Corollary 4.5
For \(p \ge 19,\) the Galois representation \(\overline{\rho }\) is surjective.
Proof
In the proof of [12, Lemma 3.7], the authors used that \(v_{{\mathfrak {a}}}(j(E)) <0\) and \(p \not \mid v_{{\mathfrak {a}}}(j(E))\) to deduce that E has multiplicative reduction at \({\mathfrak {a}} \in S\) and that the cardinality of \(\overline{\rho }(I_{{\mathfrak {a}}})\) is also divisible by p, where \(I_{{\mathfrak {a}}}\) is the inertia subgroup at \({\mathfrak {a}}.\) Since all the irreducible subgroups of \({{\mathrm{GL_2}}}({\mathbb {F}}_p)\) that contain an element of order p have \({{\mathrm{SL_2}}}({\mathbb {F}}_p)\) as a subgroup, we get that \({{\mathrm{SL_2}}}({\mathbb {F}}_p) \subseteq \overline{\rho }(G_K).\) The determinant of \({\overline{\rho }}\) is the mod p cyclotomic character \(\chi _p,\) and since \(K \cap {\mathbb {Q}}(\zeta _p) = {\mathbb {Q}},\) the latter is surjective. All the above imply that \(\overline{\rho }(G_K)= {{\mathrm{GL_2}}}({\mathbb {F}}_p).\) \(\square \)
5 Applying Serre’s conjecture
The trace elements \(Tr(\overline{\rho }({{\mathrm{Frob}}}_{(\pi )}) )\) lie in \({\mathbb {F}}_p,\) therefore \(\Phi \) corresponds to a class in \(H^1(Y_0({\mathcal {N}}),\, {\mathbb {F}}_p)\) that is an eigenvector for all such Hecke operators \(T_{\pi }.\)
As previously described, the obstruction in lifting such mod p eigenforms to complex eigenforms is given by the presence of ptorsion in \(H^2(Y_0({\mathcal {N}}),\, {\mathbb {Z}}).\) It is known (see for example [2, p. 202]) that if the last common multiple of the orders of elements of finite order in \(\Gamma _0({\mathcal {N}})\) is invertible in the coefficients module, then simplicial cohomology and group cohomology are the same, in other words \(H^2\left( Y_0({\mathcal {N}}),\, {\mathbb {Z}}\left[ \frac{1}{6}\right] \right) \cong H^2\left( \Gamma _0({\mathcal {N}}),\, {\mathbb {Z}}\left[ \frac{1}{6}\right] \right) .\)
Lefschetz duality for cohomology with compact support [26, Sect. 2] gives a relation between the first homology and the second cohomology \(H_{1}\left( \Gamma _0({\mathcal {N}}),\, {\mathbb {Z}}\left[ \frac{1}{6}\right] \right) \cong H^2\left( \Gamma _0({\mathcal {N}}) ,\, {\mathbb {Z}} \left[ \frac{1}{6}\right] \right) .\) It is also known that the abelianization \(\Gamma _0({\mathcal {N}})^{ab} \cong H_{1}(\Gamma _0({\mathcal {N}}),\, {\mathbb {Z}})\) and therefore, for primes \(p >3,\) if the group \(H^2(Y_0({\mathcal {N}}) ,\, {\mathbb {Z}})\) has a ptorsion element, then \(\Gamma _0({\mathcal {N}})^{ab}\) will have a ptorsion as well.
Using the Magma implementation, kindly provided to us by Haluk Şengün, of his algorithm in [26] we computed the abelianizations \(\Gamma _0({\mathcal {N}})^{ab}.\) This algorithm uses as input generators and relations for \({{\mathrm{GL_2}}}({\mathcal {O}}_K)\) which were computed by Swan [32]. The reader can consult the Magma code at:
https://warwick.ac.uk/fac/sci/maths/people/staff/turcas/fermatprog.
Prime torsion in \(\Gamma _0({\mathcal {N}})^{ab}\)
Number field  Level \({\mathcal {N}}\)  Primes l such that \(\Gamma _0({\mathcal {N}})^{ab}[l] \ne 0\) 

\({\mathbb {Q}}(i)\)  \({\mathfrak {a}}^8\)  2 
\({\mathbb {Q}}(\sqrt{2})\)  \({\mathfrak {a}}^8\)  2 
\({\mathbb {Q}}(\sqrt{7})\)  \({\mathfrak {a}}_1^4 {\mathfrak {a}}_2\)  2, 3 
\({\mathfrak {a}}_1 {\mathfrak {a}}_2^4\)  2,3 
Since we have chosen \(p \ge 19\) and there is no ptorsion in the subgroups of interest, the mod p eigenforms must lift to complex ones. Using the available Magma package for Bianchi modular forms, we compute these spaces of eigenforms. The implementation, due to Dan Yasaki, is based on an algorithm of Gunnels [16] for computing the Voronoi polyhedron and provides a replacement for the modular symbol algorithm used by Cremona to compute the action of the Hecke operators in [6].
The dimension of the respective cuspidal spaces for \({\mathbb {Q}}(i),\, {\mathbb {Q}}(\sqrt{7})\) is 0 and the dimension of the cuspidal space of level \({\mathfrak {a}}^8\) for \({\mathbb {Q}}(\sqrt{2})\) is 6.
Since there are no eigenforms at the predicted levels for \(K=\mathbb Q(i)\) or \(K= {\mathbb {Q}}(\sqrt{7}),\) the proof of Theorem 1.1 in those cases is complete.
Let us now focus on \(K= {\mathbb {Q}}(\sqrt{2}).\) As previously noted, the total dimension of the cuspidal space is 6, there are only 2 newforms at this level. On the levels \({\mathfrak {a}}^{i},\) \(1 \le i \le 7\) there is only newform at level \({\mathfrak {a}}^5,\) where \({\mathfrak {a}}\) is the only prime ideal above 2. All these forms have integer Hecke eigenvalues.
At the moment of writing this preprint LMFDB [25] did not include a database for Bianchi modular forms, so the author computed the eigenvalues independently using the available Magma packages. The results match perfectly with the ones that were since then made available in the LMFDB by John Cremona.
In the spirit of Langland’s philosophy, there should be a motif attached to the cuspidal newforms above. This motif is not always an elliptic curve, as one can see in [7, Theorem 5] and [12, Sect. 4]. We prove Theorem 1.1 here by first proving that these newforms correspond to elliptic curves, but as the referees pointed out, one could finish now the proof of this theorem via the following remark.
Remark 5.1
The Bianchi newforms above do not depend on the exponent p nor on the unknown solution \((a,\,b,\,c) \in {\mathcal {O}}_K^3\) of (3.1). If one of them comes from a solution to (3.1) via the above discussion, then its Hecke eigenvalue at \((\sqrt{2}1) \subseteq {\mathcal {O}}_K\) is congruent modulo p to \({{\mathrm{Tr}}}\overline{\rho }_{E,a,b,c}({{\mathrm{Frob}}}_{(\sqrt{2}1)})\) and one obtains an upper bound on p which completes the proof of Theorem 1.1. The reader interested mostly in the main result can directly consult the relevant numerical values on the last page of this article.
We say that an elliptic curve C / K of conductor \({\mathcal {N}} \subseteq {\mathcal {O}}_K,\) corresponds to a cuspidal Bianchi modular form F for \(\Gamma _0({\mathcal {N}})\) if the Lseries \(L(C/K,\,s)\) is the Mellin transform \(L(F,\,s)\) of F.
If C / K doesn’t have complex multiplication and in addition \(L(C,\,s)\) together with its character twists have analytic continuation and satisfies the functional equation, then by results of [19] it follows that there exists a Bianchi modular form for \(\Gamma _0({\mathcal {N}}),\) that is a newform, such that its Mellin transform is equal to \(L(C,\,s).\)
Since the curves \(E_1,\, E_2\) and \(E_3\) are base changes from elliptic curves defined over \({\mathbb {Q}},\) by the celebrated modularity theorem [4], we know that their Lfunctions have analytic continuation.
In this very fortunate situation, we can establish the desired connection between these curves defined over K and the Bianchi modular forms mentioned above. Instead of using the Faltings–Serre method, we are going to use the theory of lifting classical modular forms from \({\mathbb {Q}}\) to forms of imaginary quadratic fields with discriminant. A summary of the theory in this special case is given in [7, Sect. 4].
An aspect of this theory worth mentioning is the following. Suppose that f is a cuspidal modular form (in the classical sense) such that f does not have complex multiplication by K (i.e., \(f \otimes \chi \ne f,\) where \(\chi \) is the quadratic character attached to \(K/ {\mathbb {Q}}\)). Then, the lift F of f is cusp form for K for \(\Gamma _0( {\mathfrak {N}}),\) where \({\mathfrak {N}}\) is an ideal of \({\mathcal {O}}_K\) supported only on the primes of K that divide D (the discriminant of the quadratic extension) and the level of the cusp form f we started with. If F is the lift of a newform f, then it is also the lift of \(f \otimes \chi \) and of no other cuspidal newform over \({\mathbb {Q}}.\)
Lemma 5.2
Let K be a quadratic imaginary field of discriminant \(D\) and \(\chi \) the Dirichlet character associated to \(K/{\mathbb {Q}}.\) Suppose \(f \in S_2(\Gamma _0(N_1))\) is a classical cuspidal newform of level \(N_1 \in {\mathbb {Z}}_{\ge 1}\) such that \(f \otimes \chi \in S_2(\Gamma _0(N_2))\) is classical cupidal form of level \(N_2 \in {\mathbb {Z}}_{\ge 1},\) \(f \ne f \otimes \chi .\) Then f and \(f \otimes \chi \) lift to the same form F for K with level an ideal of norm \(N_1N_2/D^2.\)
Proof
We only have to prove the statement about the norm of level of F and this is a relatively easy exercise which makes use of the functional equations.
We should remind the reader that when we say that an elliptic curve corresponds to a newform both classical or over K, we actually mean that they have matching Lfunctions.
The rational elliptic curve \(E_1/ {\mathbb {Q}}\) corresponds to the cuspidal modular form \(f_1(z) = q 2q^5  3q^9 +O(q^{10})\) of weight 2 and level 32 and coefficient field \({\mathbb {Q}}.\)
If we let \(\chi \) be the character associated to the quadratic imaginary extension \(K/ {\mathbb {Q}}\) and twist \(f_1\) by this character, we get \(f_1 \otimes \chi (z) = q +2q^5 3q^9 +O(q^{10}),\) a rational cuspidal modular form of the same weight. The level of \(f_1 \otimes \chi \) is 64 and its Lfunction matches the one of the rational elliptic curve \(E_1^{\chi }\) (the twist of \(E_1\) by \(\chi \)).
Definition 5.3
Given two elliptic curves E and \(E',\) defined over a number field K, and some rational prime number p, we write that \(E \sim _p E'\) if their corresponding mod p Galois representations \(\overline{\rho }_{E},\, \overline{\rho }_{E'}: G_K \rightarrow {{\mathrm{GL_2}}}({\mathbb {F}}_p)\) are isomorphic.
Denote by \(\overline{\rho }_{E_i}\) the mod p Galois representations associated to \(E_i\) for all \(1 \le i \le 3.\) So far, we have used Conjecture 2.1 to prove that from \((a,\,b,\,c) \in \mathcal O_K,\) a nontrivial solution to 3.1, one obtains an elliptic curve \(E_{a,b,c}.\) The mod p Galois representation of this curve has the property there exists \(i \in \{1,\,2,\,3 \}\) such that for almost all prime ideals \((\pi ) \subseteq {\mathcal {O}}_K,\) \({{\mathrm{Tr}}}\overline{\rho }_{E,a,b,c}({{\mathrm{Frob}}}_{(\pi )}) = {{\mathrm{Tr}}}\overline{\rho }_{E_i}({{\mathrm{Frob}}}_{(\pi )}).\)
Since the Frobenius elements are dense and \(\overline{\rho }_{E,a,b,c}\) is irreducible, Brauer–Nesbitt theorem gives that \(\overline{\rho }_{E,a,b,c}\) and \(\overline{\rho }_{E_i}\) are isomorphic.
It’s very easy to see that (5.2) can’t happen. This is because \(E_1\) has complex multiplication and this imposes extra restrictions on its mod p representation \(\overline{\rho }_{E_1}.\) In particular, the latter is not surjective, but we proved that \(\overline{\rho }_{E,a,b,c}\) is surjective.
Now we restrict our attention to (5.3) and (5.4). Notice that the ideal \((3) {\mathcal {O}}_K\) splits as \((3)=(\sqrt{2} + 1 )(\sqrt{2} 1).\) We denote by \({\mathfrak {m}}_1 = (\sqrt{2} + 1 ).\)
Suppose that \(E=E_{a,b,c}\) has good reduction at \({\mathfrak {m}}_1.\) The fact that E / K has full 2torsion implies that \(4  \# E({\mathcal {O}}_K / {\mathfrak {m}}_1).\) The Hasse bounds \(1 \le \#E({\mathcal {O}}_K / {\mathfrak {m}}_1) \le 7\) imply that \(\# E(\mathcal O_K / {\mathfrak {m}}_1)=4.\)
The curves \(E_2\) and \(E_3\) have good reduction at \({\mathfrak {m}}_1\) and \(\# E_2({\mathcal {O}}_K / {\mathfrak {m}}_1) = 6\) and \(\# E_3(\mathcal O_K / {\mathfrak {m}}_1)=2.\) If (5.3) holds, then \(p  \# E_2({\mathcal {O}}_K / {\mathfrak {m}}_1)  \# E({\mathcal {O}}_K / \mathfrak m_1)=2.\) Similarly, if (5.4) is true, then \(p  \# E_3({\mathcal {O}}_K / {\mathfrak {m}}_1)  \# E({\mathcal {O}}_K / \mathfrak m_1)=2.\) Both lead us to contradictions with the size of the exponent p.
Acknowlegements
We are indebted to the referees for their careful reading of this paper and for many helpful remarks. The author is very grateful to his advisor Samir Siksek for suggesting the problem and for his great support. It is also a pleasure to thank John Cremona and Haluk Şengün for useful discussions. The author is supported by an EPSRC PhD studentship.
Declarations
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