Rigid local systems, Hilbert modular forms, and Fermat’s last theorem

Henri Darmon

March 3, 1999

Contents

1 Frey representations 4 1.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.2 Classification: the rigidity method . . . . . . . . . . . . . . . . 6 1.3 Construction: hypergeometric abelian varieties . . . . . . . . . 10

2 Modularity 20 2.1 Hilbert modular forms . . . . . . . . . . . . . . . . . . . . . . 20 2.2 Modularity of hypergeometric abelian varieties . . . . . . . . . 23

3 Lowering the level 28 3.1 Ribet’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 28 3.2 Application to x^{p} + y^{p} = z^{r} . . . . . . . . . . . . . . . . . . . 29

4 Torsion points on abelian varieties 43

Introduction

Historically, two approaches have been followed to study the classical Fermat equation x^{r}+y^{r} = z^{r}. The first, based on cyclotomic fields, leads to questions about abelian extensions and class numbers of K = ℚ(ζ_{r}) and values of the Dedekind zeta-function ζ_{K}(s) at s = 0. Many open questions remain, such as Vandiver’s conjecture that r does not divide the class number of ℚ(ζ_{r})^{+}. The second approach is based on modular forms and the study of two-dimensional representations of Gal(ℚ̄/ℚ). Even though two-dimensional representations are more subtle than abelian ones, it is by this route that Fermat’s Last Theorem was finally proved. (Cf. [Fre], [Se2], [Ri2], [W3], and [TW], or [DDT] for a general overview.)

This article examines the equation

x^{p} + y^{q} = z^{r}.  (1)

Certain two-dimensional representations of Gal(K̄/K), where K is the real subfield of a cyclotomic field, emerge naturally in the study of equation (1), giving rise to a blend of the cyclotomic and modular approaches. The special values ζ_{K}(−1) – which in certain cases are related to the class numbers of totally definite quaternion algebras over K – appear as obstructions to proving that (1) has no solutions. The condition that r is a regular prime also plays a key role in the analysis leading to one of our main results about the equation x^{p} + y^{p} = z^{r} (theorem 3.22).

One is interested in primitive solutions (a, b, c) to equation (1), i.e., those satisfying gcd(a, b, c) = 1. (Such a condition is natural in light of the abc conjecture for example. See also [Da1].) A solution is called non-trivial if abc ≠ 0. It will be assumed from now on that the exponents p, q, and r are prime and that p is odd.

Let (a, b, c) be a non-trivial primitive solution to equation (1). One wishes to show that it does not exist. The program for obtaining the desired contradiction, following the argument initiated by Frey and brought to a successful conclusion by Wiles in the case of x^{p} + y^{p} = z^{p}, can be divided into four steps.

Step 1: (Frey, Serre) Associate to (a, b, c) a mod p Galois representation

ρ : Gal(K̄/K) ⟶ GL_{2}(𝔽)

having “very little ramification”: i.e., whose ramification can be bounded precisely and a priori independently of the solution (a, b, c). Here K is a number field and 𝔽 is a finite field. For the Fermat equation x^{p} + y^{p} = z^{p}, one may take K = ℚ and 𝔽 = ℤ/pℤ: the representation ρ is then obtained by considering the action of G_{ℚ} on the p-division points of the Frey elliptic curve y^{2} = x(x − a^{p})(x + b^{p}). As will be explained in section 1, one is essentially forced to take K = ℚ(ζ_{q}, ζ_{r})^{+} and 𝔽 the residue field of K at a prime above p in studying equation (1).

Step 2: (Wiles) Prove that ρ is modular, i.e., arises from a Hilbert modular form on GL_{2}(K). In the setting of Fermat’s equation, Wiles proves that all semistable elliptic curves over ℚ arise from a modular form, which implies the modularity of ρ.

Step 3: (Ribet) Assuming step 2, show that ρ comes from a modular form of small level, and deduce (in favorable circumstances) that its image is small, i.e., contained in a Borel subgroup or in the normalizer of a Cartan subgroup of GL_{2}(𝔽). In the setting of Fermat’s equation, Ribet showed that ρ has to be reducible; for reasons that will be explained in section 3, one cannot rule out the case where the image of ρ is contained in the normalizer of a Cartan subgroup when dealing with equation (1).

Step 4: (Mazur) Show that the image of ρ is large; for example, that it contains SL_{2}(𝔽). Historically, this is the step in the proof of Fermat’s Last Theorem that was carried out first, in the seminal papers [Ma1] and [Ma2] which also introduced many of the tools used in steps 2 and 3.

In the classical setting, combining the conclusions of steps 3 and 4 leads to a contradiction and shows that (a, b, c) does not exist, thus proving Fermat’s Last Theorem. In [Da2] and [DMr], it was observed that the program above can be used to show that x^{p} + y^{p} = z^{r} has no non-trivial primitive solutions when r = 2, 3 and p ≥ 6 − r. (The result for r = 3 being conditional on the Shimura-Taniyama conjecture, which is still unproved for certain elliptic curves whose conductor is divisible by 27.) The purpose of this article is to generalize the analysis to the general case of equation (1).

Sections 1, 2, 3, and 4 describe the generalizations of steps 1, 2, 3 and 4 respectively. As a concrete application, the main results of section 3 relate solutions to x^{p}+y^{p} = z^{r} to questions about p-division points of certain abelian varieties with real multiplications by ℚ(cos(2π/r)). Alas, our understanding of these questions (and of the arithmetic of Hilbert modular forms over totally real fields) is too poor to yield unconditional statements. For the time being, the methods of this paper should be envisaged as as a way of tying equation (1) to questions which are more central, concerning Galois representations, modular forms, and division points of abelian varieties.

Acknowledgements: The author is grateful to F. Diamond, J. Ellenberg, A. Kraus, and K. Ribet for their helpful comments, and to N. Katz and J.-F. Mestre for pointing out a key construction used in section 1. The author greatly benefitted from the support of CICMA and the hospitality of the Université Paris VI (Jussieu) and the Institut Henri Poincaré where the work on this paper was started, and of the ETH in Zürich where it was completed. This work was partly funded by grants from NSERC and FCAR and by an Alfred P. Sloan research award.

1 Frey representations

1.1 Definition

If K is any field of characteristic 0, write G_{K} := Gal(K̄/K) for its absolute Galois group. Typically, K will be a number field; let K(t) be the function field over K in an indeterminate t. The group G_{K(t)} fits into the exact sequence

1 ⟶ G_{K̄(t)} ⟶ G_{K(t)} ⟶ G_{K} ⟶ 1.

Let 𝔽 be a finite field, embedded in a fixed algebraic closure of its prime field.

Definition 1.1 A Frey representation associated to the equation x^{p}+y^{q} = z^{r} over K is a Galois representation

ϱ = ϱ(t) : G_{K(t)} ⟶ GL_{2}(𝔽)

satisfying 1. The restriction of ϱ to G_{K̄(t)} has trivial determinant and is irreducible.

Let

ϱ̄^{geom} : G_{K̄(t)} ⟶ PSL_{2}(𝔽)

be the projectivization of this representation.

2. The homomorphism ϱ̄^{geom} is unramified outside {0, 1, ∞}.

3. It maps the inertia groups at 0, 1 and ∞ to subgroups of PSL_{2}(𝔽) of order p, q and r respectively.

The characteristic of 𝔽 is also called the characteristic of the Frey representation.

One should think of ϱ = ϱ(t) as a one-parameter family of Galois representations of G_{K} indexed by the parameter t. Condition 1 in the definition ensures that this family has constant determinant but is otherwise “truly varying” with t. The motivation for the definition of ϱ(t) is the following:

Lemma 1.2 There exists a finite set of primes S of K depending on ϱ in an explicit way, such that, for all primitive solutions (a, b, c) to the generalized Fermat equation x^{p}+y^{q} = z^{r}, the representation ρ := ϱ(a^{p}/c^{r}) has a quadratic twist which is unramified outside S.

Sketch of proof: Let

ϱ̄: G_{K(t)} ⟶ PGL_{2}(𝔽)

be the projective representation deduced from ϱ. The field fixed by the kernel of ϱ̄ is a finite extension of K(t), whose Galois group is identified with a subgroup G of PGL_{2}(𝔽) by ϱ̄; in other words, it is the function field of a G-covering of P_{1} over K. This covering is unramified outside {0, 1, ∞} and its ramification indices are p, q and r above those three points: it is a Gcovering of “signature (p, q, r)” in the sense of [Se1], sec. 6.4. The lemma now follows from a variant of the Chevalley-Weil theorem for branched coverings. (See for example [Be] or [Da1].)

Definition 1.3 Two Frey representations ϱ_{1} and ϱ_{2} attached to equation (1) are said to be equivalent if their corresponding projective representations ϱ̄_{1} and ϱ̄_{2} differ by an inner automorphism of PGL_{2}(𝔽̄), i.e., if ϱ_{1} is conjugate (over 𝔽̄) to a central twist of ϱ_{2}.

To a Frey representation ϱ we assign a triple (σ_{0}, σ_{1}, σ_{∞}) of elements in PSL_{2}(𝔽) of orders p, q and r satisfying σ_{0}σ_{1}σ_{∞} = 1 as follows. (Cf. [Se1], ch. 6.) The element σ_{j} is defined as the image by ϱ̄^{geom} of a generator of the inertia subgroup of G_{K̄(t)} at t = j. The elements σ_{0}, σ_{1}, and σ_{∞} are well-defined up to conjugation once primitive p, q and r-th roots of unity have been chosen. One can choose the decomposition groups in such a way that the relation σ_{0}σ_{1}σ_{∞} = 1 is satisfied (cf. [Se1], th. 6.3.2.). The triple (σ_{0}, σ_{1}, σ_{∞}) is then well-defined up to conjugation.

If C_{j} is the conjugacy class of σ_{j} in PSL_{2}(𝔽), one says that the Frey representation ϱ is of type (C_{0}, C_{1}, C_{∞}).

For the following definition, assume that the exponents p, q and r are odd, so that σ_{0}, σ_{1} and σ_{∞} lift to unique elements σ̃_{0}, σ̃_{1} and σ̃_{∞} of SL_{2}(𝔽) of orders p, q and r respectively.

Definition 1.4 The Frey representation attached to x^{p} + y^{q} = z^{r} is said to be odd if σ̃_{0}σ̃_{1}σ̃_{∞} = −1, and is said to be even if σ̃_{0}σ̃_{1}σ̃_{∞} = 1.

1.2 Classification: the rigidity method

If n is an integer, let ζ_{n} denote a primitive nth root of unity. Given an odd prime p, write p^{∗} := (−1)^{(p−1)/2}p, so that ℚ(\sqrt{p^{∗}}) is the quadratic subfield of ℚ(ζ_{p}). We now turn to the classification of Frey representations, beginning with the classical Fermat equation.

The equation x^{p} + y^{p} = z^{p}

Theorem 1.5 Let p be an odd prime. There is a unique Frey representation ϱ(t) of characteristic p (up to equivalence) associated to the Fermat equation x^{p} + y^{p} = z^{p}. One may take K = ℚ and 𝔽 = 𝔽_{p}, and the representation ϱ(t) is odd.

Remark: This theorem is originally due to Hecke [He], where it is expressed as a characterization of a certain field of modular functions of level p.

Proof of theorem 1.5: Set 𝔽 = 𝔽_{p}. We begin by classifying conjugacy classes of triples σ_{0}, σ_{1} and σ_{∞} of elements of order p in PSL_{2}(𝔽) satisfying σ_{0}σ_{1}σ_{∞} = 1. There are two conjugacy classes of elements of order p in PSL_{2}(𝔽), denoted pA and pB respectively. The class pA (resp. pB) is represented by an uppertriangular unipotent matrix whose upper right-hand entry is a square (resp. a non-square). These two classes are rational over ℚ(\sqrt{p^{∗}}) in the sense of [Se1], sec. 7.1, and are interchanged by the non-trivial element in Gal(ℚ(\sqrt{p^{∗}})/ℚ) as well as by the non-trivial outer automorphism of PSL_{2}(𝔽). Lift σ_{0}, σ_{1}, and σ_{∞} to elements σ̃_{0}, σ̃_{1} and σ̃_{∞} of order p in SL_{2}(𝔽). The group SL_{2}(𝔽) acts on the space V = 𝔽^{2} of column vectors with entries in 𝔽. Since σ̃_{j} is unipotent, there are non-zero vectors v_{1} and v_{2} in V which are fixed by σ̃_{0} and σ̃_{1} respectively. Because σ_{0} and σ_{1} do not commute, the vectors v_{1} and

v_{2} form a basis for V. Scale v_{2} so that σ̃_{0} is expressed by the matrix ( \begin{matrix}1 & 1 \\ 0 & 1 \end{matrix} )

in this basis; let ( \begin{matrix}1 & 0 \\ x & 1 \end{matrix} ) be the matrix representing σ̃_{1}. Since σ̃_{∞} has trace

2, the relation σ̃_{0}σ̃_{1} = σ̃_{∞}^{−1} forces x = 0, which is impossible since σ_{1} is of order p. Hence there are no even Frey representations of characteristic p. The relation σ̃_{0}σ̃_{1} = −σ̃_{∞}^{−1} gives x = −4. Note that the resulting elements σ_{0}, σ_{1}, and σ_{∞} belong to the same conjugacy class in PSL_{2}(𝔽). It is well-known that they generate PSL_{2}(𝔽). Hence there are exactly two distinct conjugacy classes of surjective homomorphisms

ϱ̄_{A}^{geom}, ϱ̄_{B}^{geom} : G_{ℚ̄(t)} ⟶ PSL_{2}(𝔽),

of type (pA, pA, pA) and (pB, pB, pB) respectively, which are interchanged by the outer automorphism of PSL_{2}(𝔽). By the rigidity theorem of Belyi, Fried, Thompson and Matzat (cf. [Se1], sec. 7), ϱ̄_{A}^{geom} and ϱ̄_{B}^{geom} extend uniquely to homomorphisms

ϱ̄_{A}, ϱ̄_{B} : G_{ℚ(t)} ⟶ PGL_{2}(𝔽) = Aut(PSL_{2}(𝔽))

which are conjugate to each other.Thus there is at most one Frey representation ϱ attached to x^{p}+y^{p} = z^{p}, whose corresponding projective representation ϱ̄ is conjugate to ϱ̄_{A} and ϱ̄_{B}. To prove the existence of ϱ, it is necessary to show that ϱ̄_{A} (say) lifts to a linear representation G_{ℚ(t)} ⟶ GL_{2}(𝔽). The obstruction to such a lifting is given by a cohomology class c in H^{2}(ℚ(t), ±1). We note that (for j = 0, 1, and ∞) the homomorphism ϱ̄_{A} maps the decomposition group at t = j to the normalizer of σ_{j}, which is the image in PGL_{2}(𝔽_{p}) of a Borel subgroup B of upper triangular matrices. Since the inclusion 𝔽_{p}^{×} ⟶ B splits, it follows that the restrictions c_{0}, c_{1} and c_{∞} of the cohomology class c in H^{2}(ℚ((t)), ±1), H^{2}(ℚ((t − 1)), ±1) and H^{2}(ℚ((1/t)), ±1) vanish. In particular, c has trivial “residues” at t = 0, 1, ∞ in the sense of [Se3], chapter II, annexe §2. Hence c is “constant”, i.e., comes from H^{2}(ℚ, ±1) by inflation ([Se3], chapter II, annexe, §4). Therefore it vanishes, since H^{2}(ℚ, ±1) injects in H^{2}(ℚ((t)), ±1).

The equation x^{p} + y^{p} = z^{r} Let us now turn to the equation x^{p} + y^{p} = z^{r}, where r and p are distinct primes. One is faced here with the choice of considering either Frey representations of characteristic p, or of characteristic r. From now on, we adopt the convention that the prime p is always used to denote the characteristic of the Frey representation, so that the equations x^{p} + y^{p} = z^{r} and x^{r} + y^{r} = z^{p} will require seperate consideration.

The following theorem is inspired from the proof given in [Se1], prop. 7.4.3 and 7.4.4 for the case r = 2 and r = 3, the general case following from an identical argument. (See also [DMs].)

Theorem 1.6 Suppose that r and p are distinct primes and that p ≠ 2. There exists a Frey representation of characteristic p over K associated to x^{p} + y^{p} = z^{r} if and only if

1. The field 𝔽 contains the residue field of ℚ(ζ_{r})^{+} at a prime 𝔭 above p, and

2. The field K contains ℚ(ζ_{r})^{+}.

When these two conditions are satisfied, there are exactly r − 1 Frey representations up to equivalence. When r ≠ 2, exactly (r − 1)/2 of these are odd, and (r − 1)/2 are even.

Proof: For condition 3 in definition 1.1 to be satisfied, it is necessary that PSL_{2}(𝔽) contain an element of order r. This is the reason for condition 1 in theorem 1.6. Condition 2 arises from the fact that (for r ≠ 2) the (r − 1)/2 distinct conjugacy classes of elements of order r in PSL_{2}(𝔽) are rational over ℚ(ζ_{r})^{+} and are not rational over any smaller extension. Assume conversely that conditions 1 and 2 are satisfied. Let σ_{0}, σ_{1}, and σ_{∞} be chosen as in the proof of theorem 1.5, and let σ̃_{j} be the lift of σ_{j} to SL_{2}(𝔽) of order p when j = 0, 1. Finally, let σ̃_{∞} be a lift of σ_{∞} to an element of order r if r is odd, and to an element of order 4 if r = 2. Let ω̄ ∈ 𝔽 be the trace of σ̃_{∞}. When r = 2 one has ω̄ = 0, and when r is odd, ω̄ is of the form φ(ζ_{r} + ζ_{r}^{−1}) where φ is a homomorphism from ℤ[ζ_{r} + ζ_{r}^{−1}]^{+} to 𝔽. Note that there are exactly (r − 1)/2 such φ’s. One now finds, as in the proof of theorem 1.5, that (σ̃_{0}, σ̃_{1}, σ̃_{∞}) is conjugate to one of the following two triples:

(( \begin{matrix}1 & 1 \\ 0 & 1 \end{matrix} ) , ( \begin{matrix}1 & 0 \\ −(2 + ω̄) & 1 \end{matrix} ) , ( \begin{matrix}−1 & 1 \\ −2 − ω̄ & 1 + ω̄ \end{matrix} )) ,

(( \begin{matrix}1 & 1 \\ 0 & 1 \end{matrix} ) , ( \begin{matrix}1 & 0 \\ −(2 − ω̄) & 1 \end{matrix} ) , ( \begin{matrix}1 & −1 \\ 2 − ω̄ & −1 + ω̄ \end{matrix} )) .

When r = 2, these triples are equal in PSL_{2}(𝔽). When r is odd, they are distinct. An argument based on rigidity as in the proof of theorem 1.5 produces (r−1) inequivalent homomorphisms from G_{K(t)} to GL_{2}(𝔽), yielding the desired odd and even Frey representations. These Frey representations will be constructed explicitly in section 1.3 (cf. lemma 1.9 and theorem 1.10).

The equation x^{r} + y^{r} = z^{p}

Theorem 1.7 Suppose that r and p are distinct odd primes. There exists a Frey representation of characteristic p over K associated to x^{r} + y^{r} = z^{p} if and only if

1. The field 𝔽 contains the residue field of ℚ(ζ_{r})^{+} at a prime 𝔭 above p, and

2. The field K contains ℚ(ζ_{r})^{+}.

When these two conditions are satisfied, there are exactly (r − 1)(r − 2)/2 inequivalent Frey representations: (r − 1)^{2}/4 odd representations, and (r − 1)(r − 3)/4 even representations.

Although the conclusion is somewhat different, the proof of theorem 1.7 follows the same ideas as the proof of theorem 1.6. Each triple (C_{0}, C_{1}, pA), where C_{0} and C_{1} each range over the (r − 1)/2 possible conjugacy classes of elements of order r in PSL_{2}(𝔽) gives rise to a unique odd and even projective representation of G_{K(t)} of type (C_{0}, C_{1}, pA), with one caveat: there is no even representation of type (C_{0}, C_{1}, pA) when C_{0} = C_{1}.

The equation x^{p} + y^{q} = z^{r} We finally come to the general case of equation (1). Assume that the exponents p, q and r are distinct primes and that p is odd.

Theorem 1.8 There exists a Frey representation of characteristic p over K associated to x^{p} + y^{q} = z^{r} if and only if

1. The field 𝔽 contains the residue fields of ℚ(ζ_{q})^{+} and of ℚ(ζ_{r})^{+} at a prime 𝔭 above p, and

2. The field K contains ℚ(ζ_{q})^{+} and ℚ(ζ_{r})^{+}.

When these two conditions are satisfied, there are (r−1)(q−1)/2 inequivalent Frey representations over ℚ(ζ_{q}, ζ_{r})^{+}. If q, r ≠ 2, then (r−1)(q−1)/4 of these are odd, and (r − 1)(q − 1)/4 are even.

The proof is the same as for theorems 1.5, 1.6 and 1.7.

1.3 Construction: hypergeometric abelian varieties

The equation x^{p} + y^{p} = z^{p} One can construct the Frey representation ϱ(t) of theorem 1.5 explicitly, by considering the Legendre family of elliptic curves

J = J(t) : y^{2} = x(x − 1)(x − t).

It is an elliptic curve over ℚ(t) which has multiplicative reduction at t = 0 and 1, and potentially multiplicative reduction at t = ∞. The module J[p] of its p-division points is a two-dimensional 𝔽-vector space on which G_{ℚ(t)} acts linearly. The corresponding representation ϱ(t) is the Frey representation of characteristic p attached to x^{p} + y^{p} = z^{p}.

The equation x^{p} + y^{p} = z^{r} When r = 2, let C_{2}(t) be the elliptic curve over ℚ(t) given by the equation

C_{2}(t) : y^{2} = x^{3} + 2x^{2} + tx.  (2)

Lemma 1.9 The mod p Galois representation associated to C_{2} is the Frey representation associated to x^{p} + y^{p} = z^{2}.

The proof of this lemma is omitted: it follows the same ideas but is simpler than the proof of theorem 1.10 below for the case of odd r, for which all the details are given.

Suppose now that r is an odd prime. Let ω_{j} = ζ_{r}^{j} + ζ_{r}^{−j}, and write ω for ω_{1}, so that K = ℚ(ω) is the real subfield of the cyclotomic field ℚ(ζ_{r}). Let d = (r − 1)/2 be the degree of K over ℚ.

Let g(x) = ∏_{j=1}^{d}(x + ω_{j}) be the characteristic polynomial of −ω, and let f(x) be an antiderivative of ±rg(x)g(−x); for example, we will take:

f(x) = xg(x^{2} − 2) = g(−x)^{2}(x − 2) + 2 = g(x)^{2}(x + 2) − 2.

Following [TTV], consider the following hyperelliptic curves over ℚ(t) of genus d:

C_{r}^{−}(t) : y^{2} = f(x) + 2 − 4t,  (3)
C_{r}^{+}(t) : y^{2} = (x + 2)(f(x) + 2 − 4t).  (4)

Let J_{r}^{−} = J_{r}^{−}(t) and J_{r}^{+} = J_{r}^{+}(t) be their jacobians over ℚ(t).

In [TTV], Tautz, Top, and Verberkmoes show that these families of hyperelliptic curves have real multiplications by K, i.e., that

End_{ℚ̄(t)}(J_{r}^{±}) = O_{K}.  (5)

Their proof shows that the endomorphisms of J_{r}^{±} are in fact defined over K, and that the natural action of Gal(K/ℚ) on End_{K(t)}(J_{r}^{±}) and on O_{K} are compatible with the identification of equation (5) above, which is canonical. (See also [DMs].)

Fix a residue field 𝔽 of K at a prime above p, and let φ be a homomorphism of O_{K} to 𝔽. The module J_{r}^{±}[p]⊗_{φ}𝔽 is a two-dimensional 𝔽-vector space on which G_{K} acts 𝔽-linearly. By choosing an 𝔽-basis for this vector space one obtains Galois representations (depending on the choice of φ, although this dependence is supressed from the notations)

ϱ_{r}^{±}(t) : G_{K(t)} ⟶ GL_{2}(𝔽).

Theorem 1.10 The representations ϱ_{r}^{−}(t) and ϱ_{r}^{+}(t) (as φ varies over the (r − 1)/2 possible homomorphisms from O_{K} to 𝔽) are the r − 1 distinct Frey representations of characteristic p associated to x^{p} + y^{p} = z^{r}. The representations ϱ_{r}^{−} are odd, and the representations ϱ_{r}^{+} are even.

Proof: (See also [DMs], prop. 2.2, 2.3.) Observe that 1. Outside of t = 0, 1, ∞, the curves C_{r}^{±}(t) have good reduction. Hence ϱ_{r}^{±}(t) satisfies condition 2 in definition 1.1 of a Frey representation.

2. The C_{r}^{±}(t) are Mumford curves over Spec(K[[t]]) and Spec(K[[t − 1]]), i.e., the special fiber of C_{r}^{±}(t) over these bases is a union of projective lines intersecting transversally at ordinary double points. For example, replacing y by 2y + (x + 2)g(x) yields the following equation for C_{r}^{+}(t), whose special fiber is the union of two projective lines crossing at the d+1 ordinary double points (x, y) = (−2, 0), (−ω_{j}, 0):

y^{2} + (x + 2)g(x)y + t(x + 2) = 0.  (6)

Likewise, replacing y by 2y + xg(−x) gives the following equation for C_{r}^{+}(t) over Spec(K[[t − 1]]):

y^{2} + xg(−x)y + g(−x)^{2} + (x + 2)(t − 1) = 0.  (7)

Its special fiber is a projective line with the d ordinary double points (x, y) = (ω_{j}, 0). A similar analysis can be carried out for C_{r}^{−}(t). By Mumford’s theory, the Jacobians J_{r}^{±}(t) have purely toric reduction at t = 0 and t = 1, and hence ϱ_{r}^{±} maps the inertia at these points to unipotent elements of SL_{2}(𝔽).

3. The curve C_{r}^{−}(t) has a quadratic twist which acquires good reduction over K[[(1/t)^{\frac{1}{r}}]], while C_{r}^{+}(t) acquires good reduction over this base. For example, setting t̃= (1/t)^{\frac{1}{r}} and replacing x by 1/x and y by (2y +1)/x^{(r+1)/2} in equation (4) for C_{r}^{+}(t) gives the model:

y^{2} + y = x^{r} + t̃h(x, y, t̃/2),  (8)

where h is a polynomial with coefficients in ℤ. Therefore ϱ_{r}^{−} (resp. ϱ_{r}^{+}) maps the inertia at t = ∞ to an element of order 2r (resp. r) of SL_{2}(𝔽) whose image in PSL_{2}(𝔽) is of order r.

It follows from 2 and 3 that ϱ_{r}^{±}(t) satisfies condition 3 in definition 1.1.

4. A strong version of property 1 now follows from the following grouptheoretic lemma:

Lemma 1.11 Let σ_{0}, σ_{1}, and σ_{∞} be elements of PSL_{2}(𝔽) of order p, p, and r satisfying σ_{0}σ_{1}σ_{∞} = 1. Then σ_{0}, σ_{1}, and σ_{∞} generate PSL_{2}(𝔽) unless (p, r) = (3, 5) and σ̃_{0}σ̃_{1}σ̃_{∞} = −1, in which case they generate an exceptional subgroup isomorphic to A_{5} ⊂ PSL_{2}(𝔽_{9}).

Proof: Let G be the subgroup of PSL_{2}(𝔽) generated by the images of σ_{0}, σ_{1}, and σ_{∞}. The proper maximal subgroups of PSL_{2}(𝔽) are conjugate to one of the groups in the following list (Cf. for example [Hu], ch. II.8 th. 8.27)

1. The Borel subgroup of upper triangular matrices.

2. The normalizer of a Cartan subgroup.

3. A group isomorphic to PSL_{2}(𝔽′) or PGL_{2}(𝔽′) for some 𝔽′ ⊂ 𝔽.

4. One of the exceptional subgroups A_{4}, S_{4} or A_{5}.

The fact that G contains two unipotent elements that do not commute rules out the possibility that G is contained in a Borel subgroup or the normalizer of a Cartan subgroup, and the fact that it contains an element of order r rules out the groups isomorphic to PSL_{2}(𝔽′) or PGL_{2}(𝔽′). Obviously, G can be contained in one of the exceptional subgroups only if both p and r are ≤ 5, i.e., if (r, p) = (2, 3), (2, 5), (3, 5) or (5, 3). In the first three cases G is isomorphic to PSL_{2}(𝔽_{p}). (Note that PSL_{2}(𝔽_{3}) ≃ A_{4} and that PSL_{2}(𝔽_{5}) ≃ A_{5}.) When (r, p) = (5, 3) and σ̃_{0}σ̃_{1}σ̃_{∞} = −1, one checks directly that G is isomorphic to the exceptional subgroup A_{5} ⊂ PSL_{2}(𝔽_{9}).

The equation x^{r} + y^{r} = z^{p} Choose a parameter j ∈ {1, 3, 5, …, r − 2}, and define curves over the function field ℚ(t) by the equations:

X_{r,r}^{−}(t) : y^{2r} = u^{2}x^{j−2} (\frac{x − 1}{x − u})^{j+2} ,
X_{r,r}^{+}(t) : y^{r} = u^{2}x^{j−2} (\frac{x − 1}{x − u})^{j+2} ,  u = \frac{t}{t − 1}.

A role will be played in our construction by the Legendre family J(t) of elliptic curves, whose equation we write in the more convenient form:

J(t) : y^{2} = u^{2}x^{j−2} (\frac{x − 1}{x − u})^{j+2} .

These curves are equipped with the following structures.

1. A canonical action of μ_{r} on X_{r,r,}^{−} and X_{r,r,}^{+} defined by

ζ(x, y) = (x, ζy), ζ ∈ μ_{r}.

2. An involution τ on X_{r,r}^{−}, X_{r,r}^{+} and J defined by

τ(x, y) = (u/x, 1/y).

This involution has two fixed points on X_{r,r}^{+}, and no fixed points on X_{r,r}^{−} and on J.

3. Maps π : X_{r,r}^{−} ⟶ J, and π_{r} : X_{r,r}^{−} ⟶ X_{r,r}^{+} defined by

π(x, y) = (x, y^{r}); π_{r}(x, y) = (x, y^{2}).

These maps obey the rules

τζ = ζ^{−1}τ,  πζ = π, π_{r}ζ = ζ^{2}π_{r},  τπ = πτ, τπ_{r} = π_{r}τ.

Let

C_{r,r}^{±} = X_{r,r}^{±}/τ, J′ = J/τ.

The maps π and π_{r} commute with τ and hence induce maps from C_{r,r}^{−} to J′ and C_{r,r}^{+} respectively, which will be denoted by the same letters by abuse of notation. Write π^{∗} and π_{r}^{∗} for the maps between the Jacobians of J′, C_{r,r}^{+} and C_{r,r}^{−} induced by π and π_{r} respectively by contravariant functoriality. Finally let J_{r,r}^{+} denote the Jacobian of C_{r,r}^{+}, and let J_{r,r}^{−} be the quotient of the Jacobian Jac(C_{r,r}^{−}) of C_{r,r}^{−} defined by

J_{r,r}^{−} := Jac(C_{r,r}^{−})/(π^{∗}(J′) + π_{r}^{∗}(J_{r,r}^{+})).

Proposition 1.12 The abelian varieties J_{r,r}^{+} (resp. J_{r,r}^{−}) have dimension equal to (r − 1)/2 when j ∈ {1, 3, 5, …, r − 4} (resp. j ∈ {1, 3, 5, …, r − 2}). In these cases there is a natural identification

End_{K}(J_{r,r}^{±}) = O_{K}

which is compatible with the action of Gal(K/ℚ) on each side.

Proof: The computation of the dimension of J_{r,r}^{±} is a direct calculation based on the Riemann-Hurwitz formula. To study the endomorphism rings of J_{r,r}^{±}, let

η_{ζ} : X_{r,r}^{±} ⟶ C_{r,r}^{±} × C_{r,r}^{±}

be the correspondence from C_{r,r}^{±} to C_{r,r}^{±} given by η_{ζ} := (pr, pr ∘ ζ), where pr is the natural projection of X_{r,r}^{±} to C_{r,r}^{±}. The resulting endomorphism η_{ζ} of Pic(C_{r,r}^{±}) is defined (on effective divisors) by the equation:

η_{ζ}(prP) = pr(ζP) + pr(ζ^{−1}P).

The commutation relations between ζ and π and π_{r} show that

πη_{ζ} = 2π, π_{r}η_{ζ} = η_{ζ^{2}}π_{r}.

Hence the subvarieties π^{∗}(J′) and π_{r}^{∗}(J_{r,r}^{+}) of Jac(C_{r,r}^{−}) are preserved by these correspondances, which induce endomorphisms of J_{r,r}^{−} as well as of J_{r,r}^{+}. The assignment ζ ↦ η_{ζ} yields an inclusion of O_{K} into End(J_{r,r}^{±}), which is an isomorphism since J_{r,r}^{±} is not of CM type. The result follows.

Choose as before a homomorphism φ : O_{K} ⟶ 𝔽 and let ϱ_{r,r}^{±} be the Galois representations obtained from the action of G_{K(t)} on the modules J_{r,r}^{±}[p]⊗_{φ} 𝔽. Note that the representations ϱ_{r,r}^{±} depend on the choice of the parameter j as well as on the choice of φ.

Theorem 1.13 1. The representations ϱ_{r,r}^{−}, as j ranges over {1, 3, …, r−2} and φ over the different homomorphisms O_{K} ⟶ 𝔽, are the (r−1)^{2}/4 distinct odd Frey representations attached to x^{r} + y^{r} = z^{p}.

2. The representations ϱ_{r,r}^{+}, as j ranges over {1, 3, …, r − 4} and φ over the different homomorphisms O_{K} ⟶ 𝔽, are the (r − 1)(r − 3)/4 distinct even Frey representations attached to x^{r} + y^{r} = z^{p}.

Proof: See for example [Ka], th. 5.4.4, or [CW].

Remarks:

1. The periods of the abelian varieties J_{r,r}^{±}, as functions of the variable t, are values of certain classical hypergeometric functions. These functions arise as solutions of a second-order differential equation having only regular singularities at t = 0, 1, and ∞ and monodromies of order r at 0 and 1 and quasi-unipotent monodromy (with eigenvalue −1 for the odd Frey representation, and 1 for the even Frey representation) at t = ∞.

2. Katz’s proof, which is based on his analysis of the behaviour of the local monodromy of sheaves under the operation of “convolution on G_{m}”, is significantly more general than the rank 2 case is used in our application. It also gives a motivic construction of rigid local systems over P^{1} − {0, 1, ∞} of any rank. Katz’s “hypergeometric motives” suggest the possibility of connecting equation (1) to higher-dimensional Galois representations, for which questions of modularity are less well understood.

3. In computing finer information such as the conductors of the Frey representations ϱ_{r,r}^{±}(a^{r}/c^{p}) at the “bad primes”, it may be desirable to have a direct proof of theorem 1.13 along the lines of the proof of theorem 1.10. The details, which are omitted, will be given in [DK].

The equation x^{p} + y^{q} = z^{r} The notion of “hypergeometric abelian variety” explained in [Ka], th. 5.4.4 and [CW], sec. 3.3 also yields a construction of the (r − 1)(q − 1)/2 Frey representations of characteristic p over K = ℚ(ζ_{q}, ζ_{r})^{+} associated to x^{p}+y^{q} = z^{r}, when p, q, r are distinct primes and p is odd. We will not describe the construction here, referring instead to [Ka], sec. 5.4 for the details. All that will be used in the sequel is the following theorem:

Theorem 1.14 If q, r ≠ 2 (resp. q = 2), there exist abelian varieties J_{q,r}^{−} and J_{q,r}^{+} (resp. J_{2,r}) over ℚ(t) of dimension (r − 1)(q − 1)/2 satisfying

End_{K}(J_{q,r}^{±}) = O_{K}, (resp. End(J_{2,r}) = O_{K})

whose mod p representations give rise to all the Frey representations in characteristic p associated to x^{p} + y^{q} = z^{r}. More precisely, fix a residue field 𝔽 of K at p, and let φ be a homomorphism of ℚ(ζ_{q})^{+}ℚ(ζ_{r})^{+} to 𝔽. There are (r − 1)(q − 1)/4 (resp. (r − 1)/2) such φ’s. Extending φ to a homomorphism O_{K} ⟶ 𝔽, let ϱ_{q,r}^{±} (resp. ϱ_{2,r}) be the Galois representation obtained from the action of G_{K(t)} on J_{q,r}^{±}[p] ⊗_{φ} 𝔽 (resp. J_{2,r}[p] ⊗_{φ} 𝔽). Then the representations ϱ_{q,r}^{±} (resp. ϱ_{2,r}) are the distinct Frey representations of characteristic p attached to x^{p} + y^{q} = z^{r}. The representations ϱ_{q,r}^{−} (resp. ϱ_{q,r}^{+}) are odd (resp. even).

Frey abelian varieties We may now assign to each solution (a, b, c) of equation (1) a “Frey abelian variety”, obtained as a suitable quadratic twist of the abelian variety J(a^{p}/b^{p}) for x^{p} + y^{p} = z^{p}, J_{r}^{±}(a^{p}/c^{r}) for x^{p} + y^{p} = c^{r}, J_{r,r}^{±}(a^{r}/c^{p}) for x^{r} + y^{r} = z^{p}, and J_{q,r}^{±}(a^{p}/c^{r}) for x^{p} + y^{q} = z^{r}. These twist are chosen in such a way as to make the corresponding mod p representations as “little ramified” as possible, in accord with lemma 1.2.

The equation x^{p} + y^{p} = z^{p} If (a, b, c) is a solution to the Fermat equation x^{p} + y^{q} = z^{r}, the elliptic curve J(a^{p}/c^{p}) has equation y^{2} = x(x − 1)(x − a^{p}/c^{p}), which is a quadratic twist (over ℚ(\sqrt{c})) of the familiar Frey curve

J(a, b, c) : y^{2} = x(x + a^{p})(x − b^{p}).

Let ρ be the associated mod p representation of G_{ℚ}.

The equation x^{p} + y^{p} = z^{r} When r = 2, we associate to a solution (a, b, c) of equation (1) the following twist of C_{2}(a^{p}/c^{2})):

C_{2}(a, b, c) : y^{2} = x^{3} + 2cx^{2} + a^{p}x.  (9)

When r is odd, the Frey hyperelliptic curves C_{r}^{−}(a, b, c) and C_{r}^{+}(a, b, c) are given by the equations

C_{r}^{−}(a, b, c) : y^{2} = c^{r}f(x/c) − 2(a^{p} − b^{p}),  (10)
C_{r}^{+}(a, b, c) : y^{2} = (x + 2c)(c^{r}f(x/c) − 2(a^{p} − b^{p})).  (11)

Note that C_{r}^{−}(a, b, c) is a non-trivial quadratic twist of C_{r}^{−}(a^{p}/c^{r}) (over the field ℚ(\sqrt{c})), while C_{r}^{+}(a, b, c) is isomorphic to C_{r}^{+}(a^{p}/c^{r}) over ℚ.

Here are the equations of C_{r}^{−}(a, b, c) for the first few values of r:

r = 3 : y^{2} = x^{3} − 3c^{2}x − 2(a^{p} − b^{p}).
r = 5 : y^{2} = x^{5} − 5c^{2}x^{3} + 5c^{4}x − 2(a^{p} − b^{p}).
r = 7 : y^{2} = x^{7} − 7c^{2}x^{5} + 14c^{4}x^{3} − 7c^{6}x − 2(a^{p} − b^{p}).

Let J_{r}^{±}(a, b, c) be the Jacobian of C_{r}^{±}(a, b, c), and let ρ_{r}^{±} be the corresponding mod p Galois representations (which depend, as always, on the choice of a homomorphism φ from O_{K} to 𝔽). The representation ρ_{r}^{±} is a quadratic twist of ϱ_{r}^{±}(a^{p}/c^{r}).

We do not write down the equations for J_{r,r}^{±}(a, b, c) or J_{q,r}^{±}(a, b, c), as we will have no further use for them in this article. A more careful study of the Frey abelian varieties J_{r,r}^{±}(a, b, c) associated to x^{r} + y^{r} = z^{p} is carried out in [DK].

Conductors We say that a Galois representation ρ : G_{K} ⟶ GL_{2}(𝔽) is finite at a prime λ if its restriction to a decomposition group at λ comes from the Galois action on the points of a finite flat group scheme over O_{K,λ}. When ℓ ≠ p, this is equivalent to ρ being unramified. Let N(ρ) denote the conductor of ρ, as defined for example in [DDT]. In particular, N(ρ) is divisible precisely by the primes for which ρ is not finite.

The equation x^{p} + y^{p} = z^{p} By interchanging a, b and c and changing their signs if necessary so that a is even and b ≡ 3 (mod 4), one finds that the conductor of ρ := ϱ(a, b, c) is equal to 2 (cf. [Se2]). The presence of the extraneous prime 2 in the conductor (in spite of the fact that all the exponents involved in the Fermat equation are odd) can be explained by the fact that the Frey representation used to construct ρ is odd, so that one of the monodromies of ϱ(t) is necessarily of order 2p. In contrast, we will see that the Galois representations obtained from even Frey representations are unramified at 2.

The equation x^{p} + y^{p} = z^{r} Let 𝔯 = (2 − ω) be the (unique) prime ideal of K above r.

Proposition 1.15 1. The representation ρ_{r}^{−} is finite away from 𝔯 and the primes above 2.

2. The representation ρ_{r}^{+} is finite away from 𝔯.

Proof: The discriminants Δ^{±} of the polynomials used in equations (10) and (11) to define C_{r}^{±}(a, b, c) are

Δ^{−} = (−1)^{\frac{r−1}{2}} 2^{2(r−1)}r^{r}(ab)^{\frac{r−1}{2} p}; Δ^{+} = (−1)^{\frac{r+1}{2}} 2^{2(r+1)}r^{r}a^{\frac{r+3}{2} p}b^{\frac{r−1}{2} p}.

If ℓ is a prime which does not divide Δ^{±}, then C_{r}^{±}(a, b, c) has good reduction at ℓ and hence ρ_{r}^{±} is finite at all primes above ℓ. So it is enough to consider the primes which divide 2ab. Suppose first that ℓ ≠ 2 divides a, and let λ denote any prime of K above ℓ. Let K_{λ} be the completion of K at λ and O_{λ} its ring of integers, and denote by ρ_{r,λ}^{±} the restriction of ρ_{r}^{±} to an inertia group I_{λ} ⊂ Gal(K̄_{λ}/K_{λ}) at λ. We observe that ρ_{r,λ}^{±} = ϱ_{r}^{±}(a^{p}/c^{r})|_{I_{λ}}, since ℓ does not divide c. To study ρ_{r,λ}^{±}, we consider the curve C_{r}^{±} over Spec(O_{λ}[[t]]). From the proof of theorem 1.10, one knows that C_{r}^{±} is a Mumford curve over K_{λ}[[t]]. Hence its Jacobian J_{r}^{±} is equipped with a (t)-adic analytic uniformization

1 ⟶ Q ⟶ T ⟶ J_{r}^{±}(K_{λ}((t))) ⟶ 1,

where T ≃ (K_{λ}((t))^{×})^{d} is a torus and Q is the sublattice of multiplicative periods. Because J_{r}^{±}(t) extends to an abelian scheme over Spec(O_{λ}[t, 1/t, 1/(t − 1)]), the lattice Q is generated by power series q_{1}, …, q_{d} in K_{λ}[[t]] whose leading coefficients α_{1}, …, α_{d} belong to O_{λ}^{×}. It follows that the extension of K_{λ} cut out by ρ_{r,λ}^{±} is contained in K_{λ}(ζ_{p}, α_{1}^{1/p}, …, α_{d}^{1/p}, t^{1/p}) where t = a^{p}/c^{r}. Since ord_{ℓ}(t) ≡ 0 (mod p), this extension is unramified at λ when ℓ ≠ p, and comes from a finite flat group scheme over O_{λ} when ℓ = p. The proof when ℓ ≠ 2 divides b proceeds in an identical manner, considering this time C_{r}^{±}(t) over Spec(O_{λ}[[(t − 1)]]) and using the fact that ord_{ℓ}(\frac{a^{p}}{c^{r}} − 1) = ord_{ℓ}(\frac{−b^{p}}{c^{r}}) ≡ 0 (mod p) to conclude. Consider finally the case where ℓ = 2. If 2 does not divide ab, then c is even. Making the substitution (x, y) = (1/u, (2v + 1)/u^{(r+1)/2}), the equation of C_{r}^{+}(a, b, c) becomes

v^{2} + v = 4c(a^{p} − b^{p})u^{r+1} − \frac{(a^{p} − b^{p})}{2}u^{r} + (lower order terms in u).

The coefficients involved in this equation are integral at 2, and \frac{a^{p}−b^{p}}{2} is odd; hence C_{r}^{+}(a, b, c) has good reduction at 2, and therefore ρ_{r}^{+} is unramified at λ. If 2 divides ab, suppose without loss of generality that it divides a. Then the equation (6) for C_{r}^{+}(t) also shows that C_{r}^{+}(a, b, c) is a Mumford curve over K_{λ}, and the result follows by the same analysis as above.

Remark: The reader will find in [Ell] a more general criterion for the Galois representations arising from division points of Hilbert-Blumenthal abelian varieties to be unramified, which relies on Mumford’s theory in an analogous way.

Proposition 1.15 implies that the conductor of ρ_{r}^{+} is a power of 𝔯, and that the conductor of ρ_{r}^{−} is divisible only by 𝔯 and by primes above 2. We now study the exponent of 𝔯 that appears in these conductors.

Proposition 1.16 1. If r divides ab, then the conductor of ρ_{r}^{−} and ρ_{r}^{+} at 𝔯 divides 𝔯.

2. If r does not divide ab, then the conductor of ρ_{r}^{−} and ρ_{r}^{+} at 𝔯 divides 𝔯^{3}.

Proof: We treat the case of ρ_{r}^{+}, since the calculations for ρ_{r}^{−} are similar. By making the change of variable x = (2 − ω)u − 2, y = (2 − ω)^{d+1}v in equation (6) one finds the new equation for C_{r}^{+}:

C_{r}^{+} : v^{2} + u ∏_{j}(u − \frac{2 − ω_{j}}{2 − ω})v + \frac{t}{(2 − ω)^{d}} u = 0.  (12)

Setting t̃= \frac{t}{(2−ω)^{d}}, one sees that C_{r}^{+}(t̃) is a Mumford curve over Spec(O_{𝔯}[[t̃]]). (The singular points in the special fiber have coordinates given by (u, v) = (0, 0) and (\frac{2−ω_{j}}{2−ω} , 0), which are distinct since \frac{2−ω_{j}}{2−ω} ≡ j^{2} (mod 𝔯).) One concludes that when ord_{𝔯}(t) > d, then the representation ϱ_{r}^{+}(t) is ordinary at 𝔯, and its conductor divides 𝔯. When r divides a one has ord_{𝔯}(a^{p}/c^{r}) ≥ pd > d. A similar reasoning works when r divides b, and so part 1 of proposition 1.16 follows.

Part 2 is proved by analyzing J_{r}^{±}(t) over Spec(O_{𝔯}[t, 1/(1 − t), 1/t]). The conductor of J_{r}^{±} over this base is constant, and one finds that the conductor of ρ_{r}^{±} is equal to 𝔯^{3}.

By combining the analysis of propositions 1.15 and 1.16, we have shown:

Theorem 1.17 1. The conductor of ρ_{r}^{−} is of the form 2^{u}𝔯^{v}, where u = 1 if ab is even. One has v = 1 if r divides ab, and v ≤ 3 otherwise.

2. The conductor of ρ_{r}^{+} divides 𝔯 if r divides ab, and 𝔯^{3} otherwise.

2 Modularity

2.1 Hilbert modular forms

Let K be a totally real field of degree d > 1, and let ψ_{1}, …, ψ_{d} be the distinct real embeddings of K. They determine an embedding of the group

Γ = SL_{2}(O_{K}) into SL_{2}(ℝ)^{d} by sending a matrix ( \begin{matrix}a & b \\ c & d \end{matrix} ) to the d-tuple

(( \begin{matrix}a_{i} & b_{i} \\ c_{i} & d_{i} \end{matrix} ))_{i=1}^{d} where a_{j} = ψ_{j}(a) and likewise for b_{j}, c_{j} and d_{j}. Through

this embedding, the group Γ acts on the product H^{d} of d copies of the complex upper half plane by Möbius transformations. More precisely, if τ = (τ_{1}, …, τ_{d}) belongs to H^{d}, then

Mτ := (\frac{a_{i}τ_{i} + b_{i}}{c_{i}τ_{i} + d_{i}} )_{i=1}^{d} .

If f is a holomorphic function on H^{d} and γ ∈ GL_{2}(K) we define

(f|_{2}γ)(τ) = det(γ) ∏(c_{i}τ_{i} + d_{i})^{−2}f(γτ).

Let Γ be a discrete subgroup of GL_{2}(K).

Definition 2.1 A modular form of weight 2 on Γ is a holomorphic function on H^{d} which satisfies the transformation rule

f|_{2}γ = f,

for all γ in Γ.

A function that vanishes at the cusps is called a cusp form on Γ. The space of modular forms of weight 2 on Γ is denoted M_{2}(Γ), and the space of cusp forms is denoted S_{2}(Γ).

Let 𝔫 be an ideal of K. We now introduce the space S_{2}(𝔫) of cusp forms of weight 2 level 𝔫, as in [W2], sec. 1.1. For this, choose a system 𝔠_{1}, 𝔠_{2}, …, 𝔠_{h} of representative ideals for the narrow ideal classes of K. Let 𝔡 denote the different of K, and assume that the 𝔠_{i} have been chosen relatively prime to 𝔫𝔡. Define

Γ_{i}(𝔫) := \{M = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ GL_{2}^{+}(K) | a, d ∈ O_{K}, b ∈ (𝔠_{i}𝔡)^{−1},
c ∈ 𝔠_{i}𝔡𝔫, ad − bc ∈ O_{K}^{×}}.

Definition 2.2 A cusp form of weight 2 and level 𝔫 is an h-tuple of functions (f_{1}, …, f_{h}) where f_{i} ∈ S_{2}(Γ_{i}(𝔫)).

Denote by S_{2}(𝔫) the space of cusp forms of weight 2 and level 𝔫.

To the reader acquainted with the case K = ℚ, the definition of S_{2}(𝔫) may appear somewhat contrived. It becomes more natural when one considers the adelic interpretation of modular forms of level 𝔫 as a space of functions on the coset space GL_{2}(𝔸_{K})/GL_{2}(K). As in the case where K = ℚ, the space S_{2}(𝔫) is a finite-dimensional vector space and is endowed with an action of the commuting self-adjoint Hecke operators T_{𝔭} for all prime ideals 𝔭 of K which do not divide 𝔫. (Cf. [W2], sec. 1.2.)

A modular form f ∈ S_{2}(𝔫) is called an eigenform if it is a simultaneous eigenvector for these operators. In that case one denotes by a_{𝔭}(f) the eigenvalue of T_{𝔭} acting on f. Let K_{f} be the field generated by the coefficients a_{𝔭}(f). It is a finite totally real extension of ℚ. If λ is any prime of K_{f}, let K_{f,λ} be the completion of K_{f} at λ and let O_{f,λ} be its ring of integers.

Eigenforms are related to Galois representations of G_{K} thanks to the following theorem:

Theorem 2.3 Let f be an eigenform in S_{2}(𝔫). There is a compatible system of λ-adic representations

ρ_{f,λ} : G_{K} ⟶ GL_{2}(O_{f,λ})

for each prime λ of K_{f}, satisfying:

trace(ρ_{f,λ}(frob_{𝔮})) = a_{𝔮}(f), det(ρ_{f,λ}(frob_{𝔮})) = Norm(𝔮),

for all primes 𝔮 of K which do not divide 𝔫λ.

Proof: When K is of odd degree, or when K is of even degree and there is at least one finite place where f is either special or supercuspidal, this follows from work of Shimura, Jacquet-Langlands, and Carayol. (Cf. [Ca].) In this case the representation ρ_{f,λ} can be realized on the λ-adic Tate module of an abelian variety over K. (It is a factor of the Jacobian of a Shimura curve associated to a quaternion algebra over K which is split at exactly one infinite place.) In the general case the theorem is due to Wiles [W1] (for ordinary forms) and to Taylor [Tay] for all f. The constructions of [W1] and [Tay] are more indirect than those of [Ca]: they exploit congruences between modular forms to reduce to the situation that is already dealt with in [Ca], but do not realize ρ_{f,λ} on the division points of an abelian variety (or even on the étale cohomology of an algebraic variety).

Let A be an abelian variety over K with real multiplications by a field E. More precisely, one requires that E is a finite extension of ℚ whose degree is equal to the dimension of A, and that A is equipped with an inclusion:

E ⟶ End_{K}(A) ⊗ ℚ.

Following a terminology of Ribet, call A an abelian variety of GL_{2}-type over K. It gives rise to a compatible system ρ_{A,λ} of λ-adic representations of G_{K} for each prime λ of E by considering the action of G_{K} on (T_{ℓ}(A)⊗ℚ_{ℓ})⊗_{E} E_{λ}. The conductor of A is defined to be the Artin conductor of ρ_{A,λ} for any prime λ of good reduction for A. (One can show that this does not depend on the choice of λ.) The following conjecture is the natural generalization of the Shimura-Taniyama conjecture in the setting of abelian varieties of GL_{2}-type:

Conjecture 2.4 (Shimura-Taniyama) If A is an abelian variety of GL_{2}- type over K of conductor 𝔫, then there exists a Hilbert modular form f over K of weight 2 and level 𝔫 such that

ρ_{f,λ} ≃ ρ_{A,λ}

for all primes λ of E.

If A satisfies the conclusion of conjecture 2.4, one says that A is modular. Conjecture 2.4 appears to be difficult in general, even with the powerful new techniques introduced by Wiles in [W3]. In connection with equation (1), one is particularly interested in conjecture 2.4 for hypergeometric abelian varieties.

Conjecture 2.5 For all t ∈ ℚ, the hypergeometric abelian variety J(t), (resp. J_{r}^{±}(t), J_{r,r}^{±}(t), J_{q,r}^{±}(t)) attached to the equation x^{p} + y^{p} = z^{p} (resp. x^{p} + y^{p} = z^{r}, x^{r} + y^{r} = z^{p}, x^{p} + y^{q} = z^{r}) is modular over ℚ (resp. ℚ(ζ_{r})^{+}, ℚ(ζ_{r})^{+}, ℚ(ζ_{q}, ζ_{r})^{+}).

2.2 Modularity of hypergeometric abelian varieties

The modularity of J The modularity of the curves in the Legendre family J follows from Wiles’ proof of the Shimura-Taniyama conjecture for semistable elliptic curves. To prove that J is modular, Wiles begins with the fact that the mod 3 representation J[3] is modular; this follows from results of Langlands and Tunnell on base change, the key fact being that GL_{2}(𝔽_{3}) is solvable. Wiles then shows (at least when the representation J[3] is irreducible and semi-stable) that every “sufficiently well-behaved” lift of J[3] is also modular. This includes the representation arising from the 3-adic Tate module of J, and hence J itself is modular.

The modularity of J_{r}^{±} and J_{r,r}^{±} When r = 2 the abelian variety J_{2} is an elliptic curve (which arises from the universal family on X_{0}(2)) and its modularity follows from the work of Wiles and its extensions [Di1].

Likewise when r = 3, the abelian varieties J_{r}^{±} and J_{r,r}^{−} are elliptic curves, so that their modularity follows from the Shimura-Taniyama conjecture. It is still conjectural in this case, in spite of the progress made toward the Shimura-Taniyama conjecture in [Di1] and [CDT]: for many values of t, the conductors of J_{3}^{±}(t) and J_{3,3}^{−}(t) are divisible by 27.

For r > 3, the prime 3 is never split in ℚ(ζ_{r})^{+}, so that the image of the Galois representation acting on J_{r}^{±}[3] or J_{r,r}^{±}[3] is contained in a product of groups isomorphic to GL_{2}(𝔽_{3^{s}}) with s > 1. Because GL_{2}(𝔽_{3^{s}}) is not solvable when s > 1, it seems difficult to directly prove the modularity of J_{r}^{±}[3] or J_{r}^{±}[3] and use the prime 3 as in Wiles’ original strategy.

Consider instead the prime 𝔯 of norm r. Since G_{ℚ} fixes 𝔯, it acts naturally on the modules J_{r}^{±}[𝔯] and J_{r,r}^{±}[𝔯] of 𝔯-torsion points of J_{r}^{±} and J_{r,r}^{±}. Furthermore, these modules are two-dimensional 𝔽_{r}-vector spaces and the action of G_{ℚ} on them is 𝔽_{r}-linear.

Theorem 2.6 1. The modules J_{r}^{−}[𝔯] and J_{r,r}^{−}[𝔯] are isomorphic to a quadratic twist of the mod r representation associated to the Legendre family J.

2. The modules J_{r}^{+}[𝔯] and J_{r,r}^{+}[𝔯] are reducible Galois representations.

Proof: By the same arguments as in the proof of theorem 1.10 one shows that the representations attached to J_{r}^{−}[𝔯] and J_{r,r}^{−}[𝔯] (resp. J_{r}^{+}[𝔯] and J_{r,r}^{+}[𝔯]), if irreducible, are Frey representations associated to the Fermat equation x^{r} + y^{r} = z^{r} which are odd (resp. even). By theorem 1.5, there is a unique odd Frey representation (up to twisting by a quadratic character) associated to x^{r} + y^{r} = z^{r}, which is the one associated to the r-torsion points on the Legendre family J(t). Part 1 follows. Since there are no even Frey representations associated to x^{r} + y^{r} = z^{r}, the reducibility of J_{r}^{+}[𝔯] and J_{r,r}^{+}[𝔯] follows as well. (Alternately, in [DMs], prop. 2.3, an explicit 𝔯-isogeny from J_{r}^{+}(t) to J_{r}^{+}(−t) defined over K is constructed, which shows that the corresponding representation is reducible, and in fact that J_{r}^{+} has a K-rational torsion point of order r.)

Let N_{r}^{±} and N_{r,r}^{±} be the conductors of the G_{ℚ}-representations J_{r}^{±}[𝔯] and J_{r,r}^{±}[𝔯].

Corollary 2.7 The G_{ℚ}-representations J_{r}^{±}[𝔯] and J_{r,r}^{±}[𝔯] arise from a classical modular form f_{0} on Γ_{0}(N_{r}^{±}) and Γ_{0}(N_{r,r}^{±}).

Proof: Since the elliptic curve J : y^{2} = x(x−1)(x−t)is modular for all t ∈ ℚ, it is associated to a cusp form on Γ_{0}(N_{J}) where N_{J} is the conductor of J(t). The lowering the level result of Ribet [Ri2] ensures that there is a form f_{0} of level N_{r}^{−} (resp. N_{r,r}^{−}) attached to J_{r}^{−}[𝔯] (resp. J_{r,r}^{−}[𝔯]). In the case of the even Frey representations, the appropriate modular form f_{0} can be constructed directly from Eisenstein series.

Consider now the restriction of the Galois representations J_{r}^{±}[𝔯] and J_{r,r}^{±}[𝔯] to G_{K}, which we denote with the same symbol by abuse of notation.

Theorem 2.8 There are Hilbert modular forms f over K giving rise to J_{r}^{±}[𝔯] or J_{r,r}^{±}[𝔯].

Proof: This is a consequence of cyclic base change, taking f to be the base change lift of f_{0} from ℚ to K.

In light of theorem 2.8, what is needed now is a “lifting theorem” in the spirit of [TW] and [W3] for Hilbert modular forms over K, which would allow us to conclude the modularity of the 𝔯-adic Tate module of J_{r}^{±} and J_{r,r}^{±}. The methods of [TW] are quite flexible and have recently been partially extended to the context of Hilbert modular forms over totally real fields by a number of mathematicians, notably Fujiwara [Fu] and Skinner and Wiles [SW]. Certain technical difficulties prevent one from concluding the modularity of J_{r}^{±} and J_{r,r}^{±} in full generality:

1. When r does not divide ab, the 𝔯-adic Tate module of J_{r}^{±} is neither flat nor ordinary at 𝔯. One needs lifting theorems that take this into account. The work of Conrad, Diamond and Taylor [CDT] is a promising step in this direction, but many technical difficulties remain to be resolved. Even when r = 3, one cannot yet prove that the elliptic curves J_{3}^{±}(t) and J_{3,3}^{−}(t) are modular for all t ∈ ℚ.

2. The reducibility of the representation J_{r}^{+}[𝔯] may cause some technical difficulties, although the recent results of Skinner and Wiles [SW] go a long way toward resolving these difficulties in the ordinary case.

As an application of the results of [SW], we have the following theorem:

Theorem 2.9 1. If r divides ab, then the abelian varieties J_{r}^{±}(a, b, c) are modular.

2. If r divides c, then the abelian varieties J_{r,r}^{±}(a, b, c) are modular.

Proof: The abelian varieties J_{r}^{±}(a, b, c) and J_{r,r}^{±}(a, b, c) have multiplicative reduction at 𝔯, by the proof of proposition 1.16. Hence the 𝔯-adic Tate modules T_{r}^{±} and T_{r,r}^{±} of these varieties, viewed as a representation of G_{K}, are ordinary at 𝔯. Since the residual representations J_{r}^{±}[𝔯] and J_{r,r}^{±}[𝔯] come from a Hilbert modular form over K, the main theorem of [SW] asserts that this property is inherited by all of their ordinary deformations, including T_{r}^{±} and T_{r,r}^{±}. The result follows.

The modularity of J_{q,r}^{±} Let K = ℚ(ζ_{q}, ζ_{r})^{+}, and let 𝔮 be a prime of K above q. This prime is totally ramified in K/ℚ(ζ_{r})^{+}. Denote by 𝔮 also the unique prime of ℚ(ζ_{r})^{+} below 𝔮, and let 𝔽 be the common residue field of ℚ(ζ_{r})^{+} and K at 𝔮.

As in the previous section, one notes that the action of G_{K} on the module J_{q,r}^{±}[𝔮] extends to an 𝔽-linear action of G_{ℚ(ζ_{r})^{+}}.

Theorem 2.10 The module J_{q,r}^{±}[𝔮] is isomorphic to a quadratic twist of J_{r}^{±}[𝔮] as a G_{ℚ(ζ_{r})^{+}}-module.

Proof: The proof is exactly the same as the proof of theorem 2.6.

Corollary 2.11 If J_{r}^{±} is modular, then so is J_{q,r}^{±}[𝔮].

Proof: The same as for corollary 2.7 and theorem 2.8, applying this time cyclic base change from ℚ(ζ_{r})^{+} to K.

Corollaries 2.7 and 2.11 suggest an inductive strategy for establishing the modularity of J, J_{r}^{±}, J_{r,r}^{±} and J_{q,r}^{±}, combining a series of base changes with successive applications of Wiles-type lifting theorems (at the last step, for Hilbert modular forms over ℚ(ζ_{q}, ζ_{r})^{+}). This strategy, and its connections with Fermat’s equation and its variants, is summarized in the flow chart below:

Modularity of J[3] ⟵

Base change

|

|  ⟵ Wiles lifting

↓ Modularity of J

↓ Modularity of J[r] ⟶

x^{r} + y^{r} = z^{r}

|

|  ⟵ Theorem 2.6 &  Theorem 2.6
↓  base change  ↓

Modularity of J_{r}^{−}[𝔯] and J_{r,r}^{−}[𝔯]

Modularity of J_{r}^{+}[𝔯]
and J_{r,r}^{+}[𝔯]
|  |
|  ⟵  Generalized  ⟶  |
↓  Wiles lifting?  ↓

Modularity of J_{r}^{−} and J_{r,r}^{−}

Modularity of J_{r}^{+}
and J_{r,r}^{+}
↓  ↓

Modularity of J_{r}^{−}[𝔮]

and J_{r,r}^{−}[𝔮] ⟶ \begin{matrix}x^{q} + y^{q} = z^{r}, \\ x^{r} + y^{r} = z^{q} \end{matrix} ⟵ Modularity of J_{r}^{+}[𝔮] and J_{r,r}^{+}[𝔮]

|  |
|  ⟵ Theorem 2.10 & ⟶  |
↓  base change  ↓

Modularity of J_{q,r}^{−}[𝔮] Modularity of J_{q,r}^{+}[𝔮]

|  |
|  ⟵  Generalized  ⟶  |
↓  Wiles lifting?  ↓

Modularity of J_{q,r}^{−} Modularity of J_{q,r}^{+}

↓  ↓

Modularity of J_{q,r}^{−}[𝔭] ⟶ x^{p} + y^{q} = z^{r}

⟵ Modularity of J_{q,r}^{+}[𝔭]

3 Lowering the level

3.1 Ribet’s theorem

Let ρ : G_{K} ⟶ GL_{2}(𝔽) be a Galois representation of G_{K} with values in GL_{2}(𝔽) where 𝔽 is a finite field.

Definition 3.1 We say that ρ is modular if there exists a Hilbert modular form f over K and a homomorphism j : O_{f} ⟶ 𝔽 such that, for all primes 𝔮 which are unramified for ρ,

trace(ρ(frob_{𝔮})) = j(a_{𝔮}(f)).

If f can be chosen to be of weight k and level 𝔫, we say that ρ is modular of weight k and level 𝔫.

The following is a generalization of Serre’s conjectures [Se2] to totally real fields, in a simple special case.

Conjecture 3.2 Suppose that

ρ : G_{K} ⟶ GL_{2}(𝔽)

is an absolutely irreducible Galois representation, where 𝔽 is a finite field of characteristic p. Suppose also that:

1. ρ is odd, and its determinant is the cyclotomic character.

2. ρ is finite at all primes 𝔭 dividing p.

3. The conductor of ρ in the sense of [Se2] is equal to 𝔫.

Then ρ is modular of weight 2 and level 𝔫.

This conjecture also seems quite difficult. (For example, the argument in [Se2], sec. 4, th. 4 shows that conjecture 3.2 implies the generalized Shimura-Taniyama conjecture 2.4.) The following conjecture, which extends a result of Ribet [Ri2] to totally real fields, should be more approachable:

Conjecture 3.3 Suppose that ρ satisfies the assumptions of conjecture 3.2, and that it is modular of weight 2 and some level. Then ρ is modular of weight 2 and level 𝔫.

The following partial result is proved in [Ja] and [Ra], building on the methods of [Ri2]:

Theorem 3.4 Let ρ : G_{K} ⟶ GL_{2}(𝔽) be an irreducible mod p representation associated to a Hilbert cuspidal eigenform f of weight 2 and level 𝔫λ, where 𝔫, λ and p are mutually relatively prime. If [K : ℚ] is even, assume that f is either special or supercuspidal at a finite prime 𝔮 not dividing p and λ. Then if ρ is unramified at λ, ρ comes from a Hilbert cuspidal eigenform g of weight 2 and level 𝔫.

3.2 Application to x^{p} + y^{p} = z^{r}

In the remainder of this article we will focus our attention on the equation x^{p} + y^{p} = z^{r} and attack it by studying the representations ρ_{r}^{+} = ϱ^{+}(a^{p}/c^{r}) (and, towards the end, ρ_{r}^{−}) attached to the p-torsion of J_{r}^{±}(a, b, c).

Theorem 3.5 1. If r divides ab then ρ_{r}^{+} (resp. ρ_{r}^{−}) comes from a modular form of weight 2 and level dividing 𝔯 (resp. 2^{u}𝔯, for some u).

2.If r does not divide ab, assume further that J_{r}^{±}(t) is modular and that conjecture 3.3 holds for Hilbert modular forms over K. Then ρ_{r}^{+} (resp. ρ_{r}^{−}) comes from a modular form of weight 2 and level dividing 𝔯^{3} (resp. 2^{u}𝔯^{3}, for some u).

Proof: The modularity of J_{r}^{±}(a, b, c) (which when r divides ab follows from theorem 2.9) implies that ρ_{r}^{±} is modular of weight 2 and some level. By theorem 1.17, ρ_{r}^{+} has conductor dividing 𝔯 when r|ab and dividing 𝔯^{3} in general, and satisfies all the other hypotheses in conjecture 3.2; a similar statement holds for ρ_{r}^{−}. Conjecture 3.3 implies the conclusion. Note that when r|ab, the Hilbert modular form f associated to J_{r}^{+}(a, b, c) is special or supercuspidal at 𝔯, so that the hypotheses of theorem 3.4 are satisfied. Hence theorem 3.4 can be applied to prove part 1 of theorem 3.5 unconditionally.

Remark: Theorem 3.5 suggests that the analysis of the solutions (a, b, c) to x^{p} + y^{p} = z^{r} splits naturally into two cases, depending on whether or not r divides ab. The following definition is inspired by Sophie Germain’s classical terminology:

Definition 3.6 A primitive solution (a, b, c) of x^{p} + y^{p} = z^{r} is called a first case solution if r divides ab, and a second case solution otherwise.

Remark: As with Fermat’s Last Theorem, the first case seems easier to deal with than the second case. (Cf. theorem 3.22.)

Our hope is that theorem 3.5 forces the image of ρ_{r}^{±} to be small (at least for some values of r). Before pursuing this matter further, observe that equation (1) has (up to sign) three trivial solutions: (0, 1, 1), (1, 0, 1) and (1, −1, 0).

Proposition 3.7 1. If (a, b, c) = (0, 1, 1) or (1, 0, 1), then J_{r}^{+} and J_{r}^{−} have degenerate reduction, and the representations ρ_{r}^{±} are therefore reducible.

2. If (a, b, c) = (1, −1, 0), then J_{r}^{±} have complex multiplication by ℚ(ζ_{r}), and hence the image of ρ_{r}^{±} is contained in the normalizer of a Cartan subgroup of GL_{2}(𝔽).

Proof: This can be shown by a direct calculation. For example, the curve C_{r}^{+}(1, −1, 0) has equation

y^{2} = x^{r+1} − 4x.

Making the substitution (x, y) = (−1/u, (2v + 1)/u^{(r+1)/2}, one obtains the equation

v^{2} + v = u^{r},

and one recognizes this as the equation for the hyperelliptic quotient of the Fermat curve x^{r} + y^{r} = z^{r} which has complex multiplication by ℚ(ζ_{r}).

Proposition 3.7 suggests the following question:

Question 3.8 Can one show that the image of ρ_{r}^{±}(a, b, c) is necessarily contained in a Borel subgroup or in the normalizer of a Cartan subgroup of GL_{2}(𝔽)?

The case r = 2 and 3 For r = 2 (resp. r = 3) one can answer this question in the affirmative, by noting that ρ_{2}(a, b, c) (resp. ρ_{3}^{+}(a, b, c)) is modular of level dividing 32 (resp. 27). (One needs to assume the Shimura-Taniyama conjecture for r = 3.) The space of classical cusp forms of weight 2 and level 32 (resp. 27) is one-dimensional. In fact X_{0}(32) (resp. X_{0}(27)) is an elliptic curve with complex multiplication by ℚ(i) (resp. ℚ(ζ_{3})). (It is also a quotient of the Fermat curve x^{4}+y^{4} = z^{4} (resp. x^{3}+y^{3} = z^{3}).) So the Galois representations arising from non-trivial primitive solutions of x^{p} + y^{p} = z^{2} and x^{p} + y^{p} = z^{3} are either reducible or of dihedral type. This was proved in [Da2]. (See also [DMr].)

To answer question 3.8 for specific values of r > 3 and p requires a computation of all the Hilbert modular forms over K of weight 2 and level dividing 𝔯^{3}. We will limit ourselves to the simpler case where K has narrow class number one.

Remark: It is known that K has narrow class number one for all r < 100 except r = 29, when the narrow class number is equal to 8. (The author is grateful to Cornelius Greither for pointing out these facts.)

We now give a formula for the dimension of S_{2}(1) and S_{2}(𝔯^{k}), with k = 1, …, 3 under the narrow class number one assumption. To do this we need to introduce some notations:

• Recall that d = (r − 1)/2 denotes the degree of K over ℚ.

• Set δ_{2} = 2 if r ≡ 1 (mod 4) and δ_{2} = 0 if r ≡ 3 (mod 4). Likewise let δ_{3} = 2 if r ≡ 1 (mod 3) and δ_{3} = 0 if r ≡ 2 (mod 3).

• Let ζ_{K}(s) be the Dedekind zeta-function of K. The main contribution to the dimension of S_{2}(𝔯^{k}) is given by the special value ζ_{K}(−1), a rational number which can be computed from the formula:

ζ_{K}(−1) = \frac{(−1)^{d}}{12} ∏_{χ} \frac{B_{2,χ}}{2} , B_{2,χ} = \frac{1}{r} ∑_{a=1}^{r} χ(a)a^{2},

where the product is taken over all non-trivial even Dirichlet characters χ : (ℤ/rℤ)^{×}/⟨±1⟩ ⟶ ℂ^{×} of conductor r.

• Let h^{−} be the minus part of class number of ℚ(ζ_{r}). This number can be evaluated also as a product of generalized Bernouilli numbers:

h^{−} = (−1)^{d}2r ∏_{χ} \frac{B_{1,χ}}{2} , B_{1,χ} = \frac{1}{r} ∑_{a=1}^{r} χ(a)a,

where the product this time is taken over the odd Dirichlet characters of conductor r.

• Let h(a) be the class number of the quadratic extension K(\sqrt{a}), and (for d < 0) let q(a) be the index of O_{K}^{×}O_{ℚ(\sqrt{a})}^{×} in O_{K}(\sqrt{a})^{×}. One has q(a) = 1 or 2, and q(a) = 1 if r ≡ 3 (mod 4). Only the ratios h(−1)/q(−1) and h(−3)/q(−3) are involved in the formula for the dimension of S_{2}(𝔯^{k}). Let χ_{4} and χ_{3} denote the non-trivial Dirichlet character mod 4 and 3 respectively. When K has narrow class number one, these ratios are given by the formulae:

\frac{h(−1)}{q(−1)} = (−1)^{d+1} ∏_{χ} \frac{B_{1,χχ_{4}}}{2} , \frac{h(−3)}{q(−3)} = (−1)^{d+1} ∏_{χ} \frac{B_{1,χχ_{3}}}{2} ,

where the products are taken over the non-trivial even Dirichlet characters of conductor r. (Recall that

B_{1,χχ_{4}} = \frac{1}{4r} ∑_{a=1}^{4r} aχχ_{4}(a), B_{1,χχ_{3}} = \frac{1}{3r} ∑_{a=1}^{3r} aχχ_{3}(a).)

The following table lists these invariants for the first few values of r:

\begin{matrix}r & d & ζ_{K}(−1) & h^{−} & h(−1)/q(−1) & h(−3)/q(−3) \\ 5 & 2 & 1/30 & 1 & 1 & 1 \\ 7 & 3 & −1/21 & 1 & 1 & 1 \\ 11 & 5 & −20/33 & 1 & 1 & 1 \\ 13 & 6 & 152/39 & 1 & 3 & 2 \\ 17 & 8 & 18688/51 & 1 & 8 & 5 \\ 19 & 9 & −93504/19 & 1 & 19 & 9 \end{matrix}

Let

χ(𝔫) = 1 + (−1)^{d} dim(S_{2}(𝔫)).

Under the assumption that K has narrow class number one, this is the arithmetic genus of the Hilbert modular variety H^{d}/Γ_{0}(𝔫); cf. [Fr], ch. II, sec. 4, th. 4.8.

Theorem 3.9 Assume that K has narrow class number 1. Then χ(𝔯^{k}) (and hence, the dimension of S_{2}(𝔯^{k})) is given by the formula:

χ(1) = \frac{ζ_{K}(−1)}{2^{d−1}} + \frac{r − 1}{2r}h^{−} + \frac{h(−1)}{4q(−1)} + \frac{h(−3)}{3q(−3)},
χ(𝔯) = (r + 1)\frac{ζ_{K}(−1)}{2^{d−1}} + \frac{r − 1}{2r}h^{−} + δ_{2} \frac{h(−1)}{4q(−1)} + δ_{3} \frac{h(−3)}{3q(−3)},
χ(𝔯^{k}) = r^{k−1}(r + 1)\frac{ζ_{K}(−1)}{2^{d−1}} + δ_{2} \frac{h(−1)}{4q(−1)} + δ_{3} \frac{h(−3)}{3q(−3)}.

Proof: The formula for χ(1) is given in [We], theorem 1.14 and 1.15. A routine calculation then yields the formula for χ(𝔯^{k}), after noting that:

1. An elliptic fixed point of order 2 (resp. 3) on H^{d} for the action of SL_{2}(O_{K}) lifts to δ_{2} (resp. δ_{3}) elliptic fixed points on H^{d}/Γ_{0}(𝔯^{k}) for k ≥ 1.

2. An elliptic fixed point of order r lifts to a unique elliptic fixed point modulo Γ_{0}(𝔯), and there are no elliptic fixed points of order r on H^{d}/Γ_{0}(𝔯^{k}) when k > 1.

Noting that K has narrow class number one when r < 23, theorem 3.9 allows us to compute the dimensions for the relevant spaces of cusp forms:

r dim(S_{2}(1)) dim(S_{2}(𝔯)) dim(S_{2}(𝔯^{2})) dim(S_{2}(𝔯^{3}))
5  0  0  0  2
7  0  0  1  5
11  0  1  6  56
13  1  4  24  290
17  6  55  879  14895
19  12  379  7300  138790

The case r = 5 and 7: When r = 5, the action of the Hecke operators on the spaces S_{2}(𝔫) over K = ℚ(\sqrt{5}) can be calculated numerically by exploiting the Jacquet-Langlands correspondence between forms on GL_{2}(K) and on certain quaternion algebras. Let B be the (unique, up to isomorphism) totally definite quaternion algebra over K which is split at all finite places. The algebra B can be identified with the standard Hamilton quaternions over K, since 2 is inert in K:

B = \{x + yi + zj + wk, x, y, z, w ∈ ℚ(\sqrt{5})\}.

The class number of B is equal to one: the maximal orders in B are all conjugate to the ring of icosians

R = ℤ[ω, i, j, k, \frac{1}{2}(1 + i + j + k), \frac{1}{2}(i + ωj + ω̄k)],

whose unit group R^{×} is isomorphic to the binary icosahedral group of order 120. (Cf. for example [CS], ch. 8, sec. 2.1.) Let R_{𝔫} be an Eichler order of level 𝔫 in R, and write

R̂_{𝔫} := R_{𝔫} ⊗ ℤ̂,  B̂ = B ⊗ ℤ̂.

The Jacquet-Langlands correspondence shows that S_{2}(𝔫) is isomorphic as a Hecke module to the space

L^{2}(R̂_{𝔫}^{×}\B̂^{×}/B^{×}),

on which the Hecke operators act in the standard way.

The following table lists the eigenvalues of the Hecke operators T_{𝔭} acting on S_{2}(𝔯^{3}), for all the primes 𝔭 of K of norm ≤ 50. It turns out that the two eigenforms in S_{2}(𝔯^{3}) are conjugate to each other over ℚ(\sqrt{5}), so we have only displayed the eigenvalues of one of the two eigenforms.

𝔭 (2) (3) (3 − ω) (4 + ω) (4 − ω) (5 + ω) (5 − ω)
a_{𝔭}(f) 0 0 \frac{−1−5\sqrt{5}}{2} \frac{−1+5\sqrt{5}}{2} 0 0 0
𝔭  (6 + ω) (7 + 2ω) (5 − 2ω) (7 + ω) (6 − ω) (7)
a_{𝔭}(f)  0  \frac{−11+5\sqrt{5}}{2} \frac{−11−5\sqrt{5}}{2}  \frac{9+5\sqrt{5}}{2}  \frac{9−5\sqrt{5}}{2}  0

Observe that a_{𝔭}(f) = 0 for all the primes 𝔭 which are inert in the quadratic extension ℚ(ζ_{5})/ℚ(ω). This suggests that f is actually of CM type, and corresponds to an abelian variety of dimension 2 with complex multiplication by ℚ(ζ_{5}).

In fact, this can be proved: the abelian variety

J_{5}^{+}(1, −1, 0) = Jac(y^{2} + y = x^{5})

has complex multiplication by ℚ(ζ_{5}), and its Hasse-Weil L-function is a product of Hecke L-series attached to Grossencharacters of ℚ(ζ_{5}) of conductor (1 − ζ_{5})^{2}. A direct calculation shows that J_{5}^{+}(1, −1, 0) is associated to the two eigenforms in S_{2}(\sqrt{5}^{3}) over GL_{2}(ℚ(\sqrt{5})).

When r = 7, we did not carry out a numerical investigation of the Hecke eigenforms of level 𝔯^{2} and 𝔯^{3}, but this turns out to be unecessary in identifying the modular forms that arise in these levels. Let A be the (unique, up to isogeny) elliptic curve over ℚ of conductor 49, which has complex multiplication by ℚ(\sqrt{−7}). It corresponds to a cusp form over ℚ of level 49. Its base change lift to K = ℚ(cos(2π/7)) is the unique modular form of level 𝔯^{2}. The space S_{2}(𝔯^{3}) contains a two-dimensional space of old forms, and hence there are three eigenforms of level 𝔯^{3}. These must consist of the Hilbert modular forms associated to the Fermat quotient

J_{7}^{+}(1, −1, 0) : y^{2} + y = x^{7}.

So when r = 5 and 7, the spaces S_{2}(𝔯^{3}) contain only eigenforms of CM type associated to hyperelliptic Fermat quotients or CM elliptic curves. Hence:

Theorem 3.10 Let r = 5 or 7, and let (a, b, c) be a non-trivial primitive solution to the equation x^{p} + y^{p} = z^{r}, where p ≠ r is an odd prime. Let 𝔭 be any prime of K = ℚ(cos(2π/r)) above p, and write 𝔽 := O_{K}/𝔭. Then

1. If (a, b, c) is a first case solution, the mod 𝔭 representation associated to J_{r}^{+}(a, b, c) is reducible;

2. If (a, b, c) is a second case solution, assume further that J_{r}^{+}(a, b, c) is modular, and that Ribet’s lowering the level theorem (conjecture 3.3) holds for Hilbert modular forms over K. Then the mod 𝔭 representation associated to J_{r}^{+}(a, b, c) is either reducible, or its image is contained in the normalizer of a Cartan subgroup of GL_{2}(𝔽).

Following [Se4], one can use the fact that J_{r}^{+} is semistable to obtain more precise information in the first case.

Proposition 3.11 If r = 5 or 7 and (a, b, c) is a first case solution to x^{p} + y^{p} = z^{r}, then J_{r}^{+}(a, b, c) is ℚ-isogenous to an abelian variety having a rational point of order p.

Proof: Choose a prime 𝔭 of K above p, and let χ_{1} : G_{K} ⟶ 𝔽^{×} be the character giving the action of G_{K} on the K-rational one-dimensional 𝔽-vector subspace L of J_{r}^{+}[𝔭]. Let χ_{2} be the character of G_{K} describing its action on J_{r}^{+}[𝔭]/L. The local analysis in [Se4], sec. 5.4., lemme 6, shows that χ_{1} and χ_{2} are unramified outside the primes above p. Also, the set of restrictions {χ_{1}|_{I_{𝔭′}}, χ_{1}|_{I_{𝔭′}}} to an inertia group I_{𝔭′} at a prime 𝔭′ above 𝔭 is equal to {χ, 1}, where χ is the cyclotomic character giving the action of I_{𝔭′} on the pth roots of unity. (Use the corollary to prop. 13 of [Se4].) Hence one of χ_{1} or χ_{2} is everywhere unramified. (When there is a single prime of K above p, this is immediate. If p is split in K, one observes, by analyzing the image of the map O_{K}^{×} ⟶ (O_{K} ⊗ 𝔽_{p})^{×} and using class field theory, that the inertia groups at the various 𝔭′ in the maximal tamely ramified abelian extension of K unramified outside p have non-trivial intersection, and in fact are equal for all but finitely many p.) Since K has class number 1, one of χ_{1} or χ_{2} is trivial. If χ_{1} = 1, then J_{r}^{+}[𝔭] has a K-rational point whose trace gives a point of order p in J_{r}^{+}(a, b, c). If χ_{2} = 1, the module L̃ generated by the O_{K}[G_{ℚ}]-translates of L is a ℚ-rational subgroup of J_{r}^{+}[p] which is of rank one over O_{K} ⊗ 𝔽_{p}. The quotient J_{r}^{+}/L̃ has a rational point of order p.

Corollary 3.12 If ℓ is a prime satisfying ℓ < p^{1/d} −2p^{1/2d} +1, then ℓ divides ab.

Proof: If ℓ does not divide ab, then J_{r}^{+} has good reduction at ℓ and #J_{r}^{+}(𝔽_{ℓ}) < (1 + \sqrt{ℓ})^{2d} by the Weil bounds. Hence p > #J_{r}^{+}(𝔽_{ℓ}). This contradicts proposition 3.11, since the prime-to-ℓ part of the torsion subgroup of of J_{r}^{+}(ℚ) injects into J_{r}^{+}(𝔽_{ℓ}) (and likewise for any abelian variey isogenous to J_{r}^{+}).

Theorem 3.13 Suppose r = 5 or 7. There exists a constant C_{r}^{−} depending only on r such that, if p ≥ C_{r}^{−} and (a, b, c) is a first case solution to x^{p} +y^{p} = z^{r}, the Galois representation ρ_{r}^{−} is reducible. (In this case there is a quotient of J_{r}^{−}(a, b, c) over ℚ which has a rational point of order p.)

Proof: By corollary 3.12, if p is large enough then 2 divides ab, so that J_{r}^{−}(a, b, c) is semistable. The mod 𝔭 representation associated to J_{r}^{−}(a, b, c), if irreducible, is therefore equal to the mod 𝔭 rerpesentation associated to a Hilbert modular form f over K in S_{2}(2𝔯). Corollary 3.12 further implies that if ℓ is a prime of ℚ and λ a prime of K above it, we have

a_{λ}(f) ≡ norm(λ) + 1 (mod 𝔭), for all ℓ ≤ p^{1/d} − 2p^{1/2d} + 1.

For each f there is a constant C_{f}^{−} such that this statement fails whenever p > C_{f}^{−}, since the mod 𝔭 representations attached to f are irreducible for almost all 𝔭. Now take C_{r}^{−} to be the maximum of the C_{f}^{−} as f runs over the normalized eigenforms in S_{2}(2𝔯). The statement in parentheses follows by applying to J_{r}^{−} the same arguments used in the proof of proposition 3.11.

Remark: Although the statement of theorem 3.13 involves only ρ_{r}^{−}, note the crucial role played in its proof by the representation ρ_{r}^{+} via corollary 3.12. This illustrates how information gleaned from one Frey representation may sometimes be used to yield insights into a second a priori unrelated Frey representation associated to the same generalized Fermat equation.

The case r = 11 When r = 11, there is a 44-dimensional space of new forms of level 𝔯^{3}, and studying the equation x^{p} + y^{p} = z^{11} would require computing the Fourier coefficients associated to these newforms. We content ourselves with the following result, which requires only dealing with S_{2}(𝔯).

Theorem 3.14 Let (a, b, c) be a first case solution to the equation x^{p} + y^{p} = z^{11}, where p > 19 is prime, and let 𝔭 be any prime of K = ℚ(cos(2π/11)) above p Then the mod 𝔭 representation associated to J_{11}^{+}(a, b, c) is reducible, and in fact J_{11}^{+}(a, b, c) has a rational point of order p.

Proof: Let 𝔭 be any ideal of K above p, and let ρ_{𝔭} denote the mod 𝔭 representation associated to J_{11}^{+}(a, b, c). Suppose that it is irreducible. By exploiting the action of Gal(K/ℚ), it follows that ρ_{𝔭} is irreducible for all choices of 𝔭. Theorem 3.5 implies that ρ_{𝔭} is modular of level dividing 𝔯. The table above shows that the space of cusp forms of this level is one-dimensional. In fact, the unique normalized eigenform f of level 𝔯 is the base change lift to K = ℚ(cos(2π/11)) of the modular form f = η(z)^{2}η(11z)^{2} of level 11 associated to the elliptic curve X_{0}(11). Consider the prime ideal (2) of K above 2, of norm 32. Then

a_{(2)}(f) = a_{32}(f) = 8.

This implies that

a_{(2)} := a_{(2)}(J_{11}^{+}(a, b, c)) ≡ 8 (mod 𝔭)

for all primes 𝔭 above p. Taking norms, one finds:

p^{5} divides norm_{K/ℚ}(a_{(2)} − 8).

By the Weil bounds, we have

|norm_{K/ℚ}(a_{(2)} − 8)| ≤ (2\sqrt{32} + 8)^{5}.

Since p > 20 > 8(1 + \sqrt{2}), we must have a_{(2)} = 8. But this leads to a contradiction. For, if 2 divides ab, then J_{11}^{+}(a, b, c) has purely toric reduction at 2 and a_{(2)} = ±1. If ab is odd, then J_{11}^{+}(a, b, c) has good reduction at (2), and 11 divides norm(32 + 1 − a_{(2)}) = 25^{5} since J_{11}^{+}(a, b, c) has a K-rational point of order 11 (by theorem 2.6). It follows that the mod p representations associated to J_{11}^{+} are reducible. The proof of prop. 3.11 now shows that J_{11}^{+}(ℚ) has a point of order p, since ℚ(cos(2π/11)) has class number one.

Corollary 3.15 If ℓ is a prime satisfying ℓ < p^{1/5} −2p^{1/10} +1, then ℓ divides ab.

The proof of this corollary is the same as for corollary 3.12. Finally, we record:

Theorem 3.16 There exists a constant C_{11}^{−} such that, if p ≥ C_{11}^{−} and (a, b, c) is a first case solution to x^{p} + y^{p} = z^{11}, the Galois representation ρ_{11}^{−} is reducible. (In this case there is a quotient of J_{11}^{−}(a, b, c) over ℚ which has a rational point of order p.)

The proof is the same as for theorem 3.13.

The case r = 13: When r = 13 there is a unique normalized cusp form of level 1, which is the base change lift of the cusp form associated to the elliptic curve X_{1}(13). (Note that this curve acquires good reduction over ℚ(cos(2π/13)).) This modular form does not pose any obstructions to studying first case solutions to x^{p} + y^{p} = z^{13}, since the representation attached to a solution of the equation is ramified at 𝔯.

On the other hand, the two-dimensional space of new forms of level 𝔯 would have to be studied more carefully in order to understand the (first case) solutions to x^{p} + y^{p} = z^{13}. The numerical calculation of eigenforms in S_{2}(𝔯) becomes increasingly difficult as r gets larger, and it has not been carried out even for r = 13. One can go further without such explicit numerical calculations, (cf. theorem 3.22 below) by studying congruences (modulo 𝔯) for modular forms.

General r: Let ℓ be a rational prime. The ℓ-adic Tate module T_{ℓ}(J_{r}^{±}(t)) ⊗ ℚ_{ℓ} is a twodimensional K_{ℓ} := K ⊗ℚ_{ℓ}-vector space. When t is rational, the linear action of G_{K} on this vector space extends to a G_{ℚ}-action which is G_{K}-semilinear, i.e., satisfies

σ(αv) = α^{σ}σ(v), for all α ∈ K_{ℓ}, σ ∈ G_{ℚ}.

Letting a_{𝔮}(J_{r}^{±}) := trace(ρ_{J,ℓ}(frob_{𝔮})), it follows that

a_{𝔮}(J_{r}^{±})^{σ} = a_{𝔮^{σ}}(J_{r}^{±}).  (13)

This motivates the following definition:

Definition 3.17 A Hilbert modular form over K of level 𝔫 is called a ℚ-form if for all ideals 𝔮 of K which are prime to 𝔫 it satisfies the relation

a_{𝔮}(f)^{σ} = a_{𝔮^{σ}}(f), for all σ ∈ G_{ℚ}.

(In particular, this implies that the Fourier coefficients a_{𝔮}(f) belong to K.)

Equation (13) implies the following lemma, which reflects the fact that the abelian varieties J_{r}^{±}(t) with t ∈ ℚ are defined over ℚ (even though their endomorphism rings are only defined over K).

Lemma 3.18 For all t ∈ ℚ, if the abelian varieties J_{r}^{−}(t) and J_{r}^{+}(t) are modular, then they are associated to a modular ℚ-form over K.

Let f be an eigenform in S_{2}(𝔫) and let λ be a prime in the ring of Fourier coefficients O_{f}. Denote by ρ_{f,λ} the λ-adic representation associated to f by theorem 2.3 and let V be the underlying K_{f,λ}-vector space. Choose a G_{K}- stable O_{f,λ}-lattice Λ in V. The space Λ̄ := Λ/λΛ gives a two-dimensional representation ρ̄_{f,λ} for G_{K} over the residue field k_{f,λ} := O_{f,λ}/λ. In general, this representation depends on the choice of lattice, but its semi-simplification does not. One says that ρ_{f,λ} is residually irreducible if ρ̄_{f,λ} is irreducible for some (and hence all) choices of lattice Λ. Otherwise one says that ρ_{f,λ} is residually reducible. In the latter case, the semi-simplification of ρ̄_{f,λ} is a direct sum of two one-dimensional characters

χ_{1}, χ_{2} : G_{K} ⟶ k_{f,λ}^{×},

whose product is the cyclotomic character

χ : G_{K} ⟶ (ℤ/ℓℤ)^{×} ⊂ k_{f,λ}^{×}

giving the action of G_{K} on the ℓ-th roots of unity.

Let f be a ℚ-form over K in the sense of definition 3.17, so that in particular its Fourier coefficients are defined over K. We say that f is 𝔯-Eisenstein if its associated 𝔯-adic representation ρ_{f,𝔯} is residually reducible.

Proposition 3.19 There exists a constant C_{r}^{+} depending only on r such that, for any first case solution (a, b, c) to equation (1) with p > C_{r}^{+}, one of the following holds:

1. The representation ρ_{r}^{+} is reducible, or

2. it is isomorphic to the mod 𝔭 representation attached to an 𝔯-Eisenstein ℚ-form in S_{2}(𝔯).

Proof: Let g be any eigenform in S_{2}(𝔯). If g is not a ℚ-form, then there exists a prime 𝔮 of O_{g} and a σ ∈ G_{ℚ} such that a_{𝔮}(g)^{σ} ≠ a_{𝔮^{σ}}(g). If g is a ℚ-form, but is not 𝔯-Eisenstein, then there is a prime 𝔮 of K such that 𝔯 does not divide a_{𝔮}(g) − N𝔮 − 1. In either case, one has

a_{𝔮}(g) ≠ a_{𝔮}(f),

for all modular forms f which correspond to a J_{r}^{+}(t) with t ∈ ℚ. Indeed, such an f is a ℚ-form and is 𝔯-Eisenstein by theorem 2.6. If f ≡ g for some prime 𝔭 of O_{g}K above p then taking norms gives

p divides Norm_{K_{g}K/ℚ}(a_{𝔮}(g) − a_{𝔮}(f)) ≠ 0.

Let d_{g} := [K_{g} : ℚ]. Applying the Weil bounds one finds:

|Norm_{K_{g}K/ℚ}(a_{𝔮}(g) − a_{𝔮}(f))| ≤ (16Norm_{K/ℚ}(𝔮))^{(r−1)d_{g}/4},

so that

p ≤ C_{g} := (16Norm_{K/ℚ}(𝔮))^{(r−1)d_{g}/4}.

In particular, if p > C_{g}, the representation ρ_{r}^{+} is not equivalent to ρ̄_{g,𝔭} for any prime of O_{g}K above p. Now set C_{r}^{+} := max_{g} C_{g}, where the maximum is taken over all eigenforms g in S_{2}(𝔯) which are either not ℚ-forms or are not 𝔯-Eisenstein.

In light of proposition 3.19, it becomes important to understand whether there exist 𝔯-Eisenstein ℚ-forms in S_{2}(𝔯).

Proposition 3.20 Suppose that r is a regular prime. Then there are no 𝔯-Eisenstein ℚ-forms over K of level 1 or 𝔯.

Proof: Suppose on the contrary that f is a ℚ-form in S_{2}(𝔯) and that ρ_{f,𝔯} is residually reducible. Let χ_{1} and χ_{2} be the characters of G_{K} which occur in the semi-simplification of Λ̄, for some (and hence all) G_{K}-stable lattices Λ in V. Because f is a ℚ-form, it follows that χ_{1} and χ_{2} are powers of the cyclotomic character χ with values in ⟨±1⟩ ⊂ 𝔽_{r}^{×}. Furthermore, χ_{1}χ_{2} = χ. Hence we may assume without loss of generality that χ_{1} = 1 and χ_{2} = χ. By proposition (2.1) of [Ri1], there exists a G_{K}-stable lattice Λ for which

ρ̄_{f,𝔯} ≃ ( \begin{matrix}χ_{1} & Ψ_{0} \\ 0 & χ_{2} \end{matrix} ) = ( \begin{matrix}1 & Ψ_{0} \\ 0 & χ \end{matrix} ) ,  (14)

and is not semi-simple. This implies that Ψ := Ψ_{0}/χ is a non-trivial cocycle in H^{1}(K, ℤ/rℤ(−1)). Proposition 3.20 now follows from the following lemma:

Lemma 3.21 The cocycle Ψ is unramified.

Proof of lemma: The cocycle Ψ is unramified at all places v ≠ 𝔯 because v does not divide the level of f. It is also unramified at 𝔯: if f is of level 1, this is because ρ̄_{f,𝔯} comes from a finite flat group scheme over K. If f is of level 𝔯, then by theorem 2 of [W1], the restriction of the representation ρ̄_{f,𝔯} to a decomposition group D_{𝔯} at 𝔯 is of the form

ρ̄_{f,𝔯}|_{D_{𝔯}} ≃ ( \begin{matrix}χ & Ψ \\ 0 & 1 \end{matrix} ) .

But the restriction of χ to D_{𝔯} is non-trivial. Comparing the equation above to equation (14), it follows that the local representation ρ̄_{f,𝔯}|_{D_{𝔯}} splits. Therefore the cocycle Ψ is locally trivial at 𝔯. This completes the proof of lemma 3.21.

Proposition 3.20 now follows directly: the cocycle Ψ cuts out an unramified cyclic extension of ℚ(ζ_{r}) of degree r, which does not exist if r is a regular prime.

Theorem 3.22 Let r be a regular prime. Then there exists a constant C_{r}^{+} (depending only on r) such that, for all p > C_{r}^{+}, and all first case solutions (a, b, c) to x^{p} + y^{p} = z^{r}, the mod 𝔭 representation associated to J_{r}^{+}(a, b, c) is reducible.

Proof: Combine propositions 3.19 and 3.20.

Remarks:

1. The value of the constant C_{r}^{+} depends on the structure of the space of Hilbert modular forms over ℚ(cos(2π/r)) of level 𝔯. It would be possible in principle to write down a crude estimate for C_{r}^{+} by using the Chebotarev density theorem and known estimates for the size of fourier coefficients of Hilbert modular eigenforms, but we have not attempted to do this.

2. The consideration of 𝔯-Eisenstein ℚ-forms so crucial for the proof of theorem 3.22 is only likely to be of use in studying first case solutions. Indeed, there typically exist 𝔯-Eisenstein ℚ-forms on S_{2}(𝔯^{3}) – for example, the base change lifts from ℚ to K of certain r-Eisenstein forms on X_{0}(r^{2}), or (more germane to the present discussion) the form in S_{2}(𝔯^{3}) asociated to the CM abelian variety J_{r}^{+}(1, −1, 0).

3. The arguments based on 𝔯-Eisenstein ℚ-forms yield no a priori information about the Galois representations ρ_{r}^{−}, since the mod r representation attached to J_{r}^{−}(a, b, c) is irreducible. (It is isomorphic to a twist of the representation coming from the r-torsion of the Frey curve y^{2} = x(x−a^{p})(x+b^{p}), by theorem 2.6.) Nonetheless, one can still show:

Theorem 3.23 Assume further that K has class number one. Then

1. J_{r}^{+}(a, b, c) has a rational point of order p.

2. There exists a further constant C_{r}^{−} such that if p > C_{r}^{−}, the abelian variety J_{r}^{−}(a, b, c) also has a rational point of order p.

Proof: The proof of 1 is the same as for proposition 3.11, and 2 follows from the same reasoning as for proposition 3.13.

4 Torsion points on abelian varieties

Ultimately one wishes to extract a contradiction from theorems like theorems 3.10, 3.13, 3.14, 3.16, 3.22 and 3.23 by proving that when p is large enough (relative to r perhaps), the image of ρ_{r}^{±} is large - for example, that this image contains SL_{2}(𝔽); or, at the very least, that the abelian varieties J_{r}^{±}(a, b, c), when semistable, cannot contain a rational point of order p. The following folklore conjecture can be viewed as a direct generalization of a conjecture of Mazur for elliptic curves.

Conjecture 4.1 Let E be a totally real field and K a number field. There exists a constant C(K, E) depending only on K and E, such that for any abelian variety A of GL_{2}-type with End_{K}(A) ⊗ ℚ = EndK̄(A) ⊗ ℚ ≃ E, and all primes 𝔭 of E of norm greater than C(K, E), the image of the mod 𝔭 representation associated to A contains SL_{2}(𝔽).

This conjecture seems difficult: the set of abelian varieties of GL_{2}-type with End(A)⊗ℚ ≃ E is parametrized by a d-dimensional Hilbert modular variety, and very little is known about the Diophantine properties of these varieties.

When r = 2 and r = 3, one has K = E = ℚ since the representations ρ_{r}^{±} arise from elliptic curves. Much of conjecture 4.1 can be proved thanks to the ideas of Mazur [Ma1], [Ma2]:

• Theorem 8 of [Ma1] implies that the image of ρ_{r}^{±} is not contained in a Borel subgroup of GL_{2}(𝔽_{p}) when p > 5.

• A result of Momose [Mo] building on the ideas in [Ma1] implies that this image is not contained in the normalizer of a split Cartan subgoup if p > 17.

• Finally, a result of Merel and the author [DMr] implies that the image of ρ_{r}^{+} is not contained in the normalizer of a non-split Cartan subgroup. (We were unable to prove a similar result for ρ_{r}^{−}.)

Combining these results with an ad-hoc study (carried out by Bjorn Poonen [Po], using traditional descent methods) of the equations x^{p} + y^{p} = z^{r} (r = 2, 3) for small values of p yields the desired contradiction. Thus the main result of [DMr] provides an (essentially) complete analogue of Fermat’s Last Theorem for equation (1) when r = 2 or 3, which one would like to emulate for higher values of r.

Theorem 4.2 ([DMr]) 1. The equation x^{p} + y^{p} = z^{2} has no non-trivial primitive solutions when p ≥ 4.

2. Assume the Shimura-Taniyama conjecture. Then the equation x^{p}+y^{p} = z^{3} has no non-trivial primitive solutions when p ≥ 3.

Return now to the case r > 3. The following special case of conjecture 4.1, which is suficient for the applications to equation (1), seems more tractable:

Conjecture 4.3 There exists a constant B_{r} depending only on r, such that for any t ∈ ℚ, and all primes 𝔭 of K = ℚ(ζ_{r})^{+} of norm greater than B_{r}, the image of the mod 𝔭 representation of G_{K} associated to J_{r}^{±}(t) is neither contained in a Borel subgroup or in the normalizer of a Cartan subgroup of GL_{2}(𝔽).

A natural approach to this conjecture is to study the curves X_{0}^{±}(𝔭), X_{s}^{±}(𝔭) and X_{ns}^{±}(𝔭) which classify the abelian varieties J_{r}^{±}(t) with a rational subgroup, a “normalizer of split Cartan subgroup” structure, and a “normalizer of non-split Cartan subroup structure” on the 𝔭-division points, where 𝔭 is an ideal of the field K.

For the moment, we know very little about the arithmetic of these curves, except when r = 2 and r = 3 when they are closely related to classical modular curves. When r > 3, they appear as quotients of the upper half plane by certain non-arithmetic Fuchsian groups described in [CW]. We will content ourselves here with giving a formula for the genus of these curves. Let ϵ = ±1 be defined by the condition ℕ𝔭 ≡ ϵ (mod r).

Lemma 4.4 1. The genus of X_{0}^{±}(𝔭) is equal to

\frac{1}{2}(1 − \frac{1}{r} − \frac{2}{p})ℕ𝔭 − \frac{ϵ}{2}(1 − \frac{1}{r}).

2. The genus of X_{s}^{±}(𝔭) is equal to

\frac{1}{4}(1 − \frac{1}{r} − \frac{2}{p})ℕ𝔭(ℕ𝔭 + 1) − \frac{ϵ + 1}{4}(1 − \frac{1}{r}) + 1.

3. The genus of X_{ns}^{±}(𝔭) is equal to

\frac{1}{4}(1 − \frac{1}{r} − \frac{2}{p})ℕ𝔭(ℕ𝔭 − 1) + \frac{ϵ − 1}{4}(1 − \frac{1}{r}) + 1.

Example: When r = 5 and 𝔭 = (3), one finds that the curves X_{0}^{−}(3) and X_{0}^{+}(3) are of genus 1, i.e., they are elliptic curves over ℚ. A direct calculation reveals that X_{0}^{+}(3) is an elliptic curve of conductor 15, denoted by 15E in Cremona’s tables. By looking up the curve 15E twisted by ℚ(\sqrt{5}), one finds that 15E has finite Mordell-Weil group over ℚ(\sqrt{5}. Does J_{0}^{+}(𝔭) always have a non-zero quotient with finite Mordell-Weil group over ℚ(ζ_{r})^{+}, at least when 𝔭 is large enough?

References

[Atlas] J.H. Conway, R.T. Curtis, S.P. Norton, R.A. Parker and R.A. Wilson. Atlas of finite groups: maximal subgroups and ordinary characters for simple groups. New York: Clarendon Press, 1985.

[Be] S. Beckmann, On extensions of number fields obtained by specializing branched coverings. J. Reine Angew. Math. 419 (1991), 27–53.

[Ca] H. Carayol, Sur les représentations ℓ-adiques associées aux formes modulaires de Hilbert. Ann. Sci. Ecole Norm. Sup. (4) 19 (1986), no. 3, 409–468.

[CDT] B. Conrad, F. Diamond, R. Taylor, Modularity of certain potentially crystalline Galois representations, manuscript, to appear.

[CS] J.H. Conway and Sloane, Sphere packings, lattices and groups. Second edition. With additional contributions by E. Bannai, R. E. Borcherds, J. Leech, S. P. Norton, A. M. Odlyzko, R. A. Parker, L. Queen and B. B. Venkov. Grundlehren der Mathematischen Wissenschaften 290. Springer-Verlag, New York, 1993.

[CW] P. Cohen, J. Wolfart, Modular embeddings for some nonarithmetic Fuchsian groups. Acta Arith. 56 (1990), no. 2, 93–110.

[Da1] H. Darmon, Faltings plus epsilon, Wiles plus epsilon, and the generalized Fermat equation, C.R. Acad.Sci. Canada, 19 (1997), no. 1, 3–14.

[Da2] H. Darmon, The equations x^{n} +y^{n} = z^{2} and x^{n} +y^{n} = z^{3}. Internat. Math. Res. Notices 10 1993, 263–274.

[DDT] H. Darmon, F. Diamond, and R. Taylor, Fermat’s Last Theorem, Current Developments in Mathematics 1, 1995, International Press, pp. 1-157.

[DK] H. Darmon, A. Kraus, On the equations x^{r}+y^{r} = z^{p}, in preparation.

[DMr] H. Darmon, L. Merel, Winding quotients and some variants of Fermat’s Last Theorem, Journal für die Reine und Angewandte Mathematik 490 (1997), 81–100.

[DMs] H. Darmon, J-F. Mestre, Courbes hyperelliptiques à multiplications réelles et une construction de Shih, CICMA preprint; Canadian Math. Bull, to appear.

[Di1] F. Diamond, On deformation rings and Hecke rings. Annals of Math (2) 144 (1996), no. 1, 137–166.

[Di2] F. Diamond, The Taylor-Wiles construction and multiplicity one. Invent. Math. 128 (1997), no. 2, 379–391.

[Ell] J. Ellenberg, Harvard PhD. Thesis, in preparation.

[Fr] E. Freitag, Hilbert Modular Forms, Springer-Verlag, 1990.

[Fre] G. Frey, Links between solutions of A − B = C and elliptic curves, in: Number theory, Ulm 1987, Proceedings, Lecture Notes in Math. 1380, Springer-Verlag, New York, 1989, 31–62.

[Fu] K. Fujiwara, Deformation rings and Hecke algebras in the totally real case, preprint.

[He] E. Hecke, Die eindeutige Bestimmung der Modulfunktionen q-ter Stufe durch algebraische Eigenschaften, Math. Ann. 111 (1935), 293-301 (=Math. Werke, 568-576).

[Hu] Huppert, B. Endliche Gruppen. I. Die Grundlehren der Mathematischen Wissenschaften, Band 134 Springer-Verlag, Berlin-New York 1967.

[Ja] F. Jarvis, On Galois representations associated to Hilbert modular forms. J. Reine Angew. Math. 491 (1997), 199–216.

[Ka] N. Katz, Exponential sums and differential equations, Annals of Math. Studies, Princeton University Press, 1990.

[Ma1] B. Mazur, Modular curves and the Eisenstein ideal. Inst. Hautes Études Sci. Publ. Math. 47 (1977), 33–186.

[Ma2] B. Mazur, Rational isogenies of prime degree. Invent. Math. 44 (1978), no. 2, 129–162.

[Me] J.-F. Mestre, Familles de courbes hyperelliptiques à multiplications réelles. Arithmetic algebraic geometry (Texel, 1989), 193-208, Progr. Math 89 Birkhauser Boston, Boston, MA, 1991.

[Mo] F. Momose, Rational points on the modular curves X_{split}(p). Compositio Math. 52 (1984), no. 1, 115–137.

[Po] B. Poonen, Some diophantine equations of the form x^{n} + y^{n} = z^{m}, J. of Number Theory, to appear.

[Ra] A. Rajaei, PhD thesis, Princeton University, 1998.

[Ri1] K. Ribet, A modular construction of unramified p-extensions of ℚ(μ_{p}), Invent. Math. 34, 151-162 (1976).

[Ri2] K. Ribet, On modular representations of Gal(ℚ̄/ℚ) arising from modular forms, Invent. Math. 100, 431–476 (1990).

[Se1] J.-P. Serre, Topics in Galois Theory. Jones and Bartlett, 1992.

[Se2] J.-P. Serre, Sur les représentations modulaires de degré 2 de Gal(\overline{Q}/Q). Duke Math. J. 54 (1987), no. 1, 179–230.

[Se3] J.-P. Serre, Galois cohomology. (edition revised by the author). Springer-Verlag, Berlin, 1997.

[Se4] J.-P. Serre, Propriétés galoisiennes des points d’ordre fini des courbes elliptiques. Invent. Math. 15 (1972), no. 4, 259–331.

[SW] C. Skinner, A. Wiles, in progress.

[TTV] W. Tautz, J. Top, A. Verberkmoes, Explicit hyperelliptic curves with real multiplication and permutation polynomials. Canad. J. Math. 43 (1991) no. 5, 1055-1064.

[Tay] R. Taylor, On Galois representations associated to Hilbert modular forms. Invent. Math. 98 (1989), no. 2, 265–280.

[TW] R. Taylor and A. Wiles, Ring-theoretic properties of certain Hecke algebras. Ann. of Math. (2) 141 (1995), no. 3, 553–572.

[Vi] M.-F. Vigneras, Arithmétique des algèbres de quaternions. Lecture Notes in Mathematics 800. Springer, Berlin, 1980.

[We] D. Weisser, The arithmetic genus of the Hilbert modular variety and the elliptic fixed points of the Hilbert modular group, Math. Ann. 257, 9–22 (1981).

[W1] A. Wiles, On ordinary λ-adic representations associated to modular forms. Invent. Math. 94 (1988), no. 3, 529–573.

[W2] A. Wiles, On p-adic representations for totally real fields. Ann. of Math. (2) 123 (1986), no. 3, 407–456.

[W3] A. Wiles, Modular elliptic curves and Fermat’s last theorem. Ann. of Math. (2) 141 (1995), no. 3, 443–551.