Modularity of fibres in rigid local systems

Henri Darmon

May 23, 2000

1 Introduction

Let K be a totally real field embedded in a fixed algebraic closure \overline{K}, and write G_{K} := Gal(\overline{K}/K) for its absolute Galois group. Fix a prime ℓ ≠ 2, and consider an odd two-dimensional Galois representation

ρ : G_{K} ⟶ GL_{2}(E),

where E is either a finite field of characteristic ℓ or a finite extension of ℚ_{ℓ}. Assume that the restrictions of ρ to the inertia groups at the primes of K above ℓ are potentially semistable in the sense of [FM].

The representation ρ is called modular if it is associated to a Hilbert modular form on GL_{2}(K), as is explained, for example, in [W1] and [W2]. Fontaine and Mazur [FM] conjectured that this is always the case. Significant progress on this conjecture was achieved [W3] by proving particular instances of the following “lifting conjecture”:

Conjecture 1.1 Suppose that ℓ is odd and that the residual representation ρ̄ attached to ρ is modular. Then ρ itself is modular.

Conjecture 1.1 is proved in [W3] and [TW] when K = ℚ and the restriction of ρ to the decomposition groups at the primes above ℓ are semistable in the sense of [DDT], sec. 2.4. This is enough (using the primes ℓ = 3 and 5) to establish the Shimura-Taniyama conjecture for semistable elliptic curves, thanks to a key result of Langlands and Tunnell. Progressively stronger cases of conjecture 1.1 were subsequently proved by [Di], [CDT], [Fu], and [SW]; in [SW], Skinner and Wiles obtain quite general results in the context where K is any totally real field, the principal assumption being that ρ is ordinary at the primes above ℓ.

In this note we consider Galois representations which occur in “rigid families”, and establish their modularity under conjecture 1.1. This implies the modularity (over suitable real abelian extensions) of the Galois representations occuring in the cohomology of the curves

y^{n} = x^{a}(x − 1)^{b}(x − t)^{c}, t ∈ ℚ,

whose periods as a function of the parameter t are values of classical hypergeometric functions.

To state the main result precisely, denote by K(t) the field of rational functions in the indeterminate t, and let

ϱ : G_{K(t)} ⟶ GL_{2}(E)

be a two-dimensional Galois representation. For x ∈ ℙ^{1}(K̄), viewed as a place of K(t), let D_{x} ⊂ G_{K(t)} be a decomposition group at x, and write I_{x} = ℤ̂(1) for its inertia subgroup. One says that ϱ is unramified at x if its restriction to I_{x} is trivial. If in addition x belongs to ℙ^{1}(K), the restriction of ϱ to D_{x} factors through D_{x}/I_{x} = G_{K}, giving rise to a Galois representation

ϱ[x] : G_{K} ⟶ GL_{2}(E),

which can be thought of as the specialization of ϱ at t = x.

Let ϱ^{geom} be the restriction of ϱ to the subgroup

G^{geom} := Gal(\overline{K(t)}/\overline{K}(t)) ⊂ G_{K(t)}.

The representation ϱ is said to be rigid if ϱ^{geom} is unramified outside 0, 1, and ∞. (The reason for this terminology will be made clear in the next section, cf. prop. 2.4.) Choose a topological generator of ℤ̂(1) corresponding to a compatible system (ζ_{n}) of primitive n-th roots of unity. For j = 0, 1, ∞, let γ_{j} be the corresponding generator of I_{j}, and let σ_{j} = ϱ(γ_{j}) ∈ GL_{2}(E). The monodromy matrices σ_{j} depend on the choice of decomposition groups D_{j} but their conjugacy classes in GL_{2}(E) are well-defined. We will show (lemma 2.2) that the semisimplification of σ_{j} has finite order n_{j}. One can then prove (prop. 2.4) that the “field of definition” K of ϱ necessarily contains the real subfield K_{n_{j}} := ℚ(ζ_{n_{j}})^{+} of the cyclotomic field of n_{j}-th roots of unity. Conversely, ϱ has a twist which extends to a representation of G_{K_{n}(t)}, where n = n(ϱ) is the least common multiple of the n_{j}. Replace ϱ by such a twist, and K by K_{n}. Our main result is then:

Theorem 1.2 Let ϱ be a rigid representation, and assume that one of the σ_{j} is unipotent, and that 8 does not divide n = n(ϱ).

If conjecture 1.1 is true, then ϱ[x] arises from a Hilbert modular form over K_{n}, for all x ∈ ℙ^{1}(ℚ) − {0, 1, ∞}.

Acknowledgements. The author thanks Nick Katz for helpful conversations and the ETH in Zürich for its hospitality while this article was written. This research was funded by grants from NSERC and by an Alfred P. Sloan research fellowship.

2 Rigid representations

Fix a rigid representation ϱ, and keep the notations of the introduction. While the monodromy matrices σ_{j} are only defined up to conjugation, the decomposition groups D_{j} can be chosen so that the relation

σ_{0}σ_{1}σ_{∞} = 1  (1)

is satisfied (cf. for example [Se1], th. 6.3.2.) Fix such a choice from now on. A 2 × 2 matrix is called a reflection if its eigenvalues are 1 and −1.

Lemma 2.1 The matrix σ_{j} is either a reflection or an element of SL_{2}(E).

Proof: The conjugacy classes of σ_{j} are rational over the real field K in the sense of [Se1], sec. 7.1. In particular, σ_{j} is conjugate to σ_{j}^{−1}; the result follows.

If one of the σ_{j} is a reflection, then exactly two are, because of the relation det(σ_{0}σ_{1}σ_{∞}) = 1 which follows from (1). In that case the image of ϱ^{geom} is a dihedral group. We exclude this case from consideration from now on, and assume that each σ_{j} belongs to SL_{2}(E). The matrix σ_{j} is said to be quasi-unipotent if its minimal polynomial has a double root.

Lemma 2.2 The σ_{j} are either quasi-unipotent or of finite order.

Proof: Let K^{cyc} := K(ζ_{∞}) be the maximal cyclotomic extension of K, and let Ω be its Galois group, identified with a subgroup of ℤ̂^{×}. Since the conjugacy class of σ_{j} is rational over K, the matrix σ_{j} is conjugate to σ_{j}^{α} for all α ∈ Ω. But Ω has finite index in ℤ̂^{×}, and hence the eigenvalues of σ_{j} are roots of unity.

Definition 2.3 An admissible triple in SL_{2}(E)) is a triple (σ_{0}, σ_{1}, σ_{∞}) of elements in SL_{2}(E), taken modulo conjugation in GL_{2}(E), and satisfying (a) The semisimplification of σ_{j} has finite order n_{j};

(b) The group generated by σ_{0}, σ_{1}, and σ_{∞} is an irreducible subgroup of SL_{2}(E).

(c) σ_{0}σ_{1}σ_{∞} = 1.

Let n = n(ϱ) = lcm(n_{0}, n_{1}, n_{∞}), as before. The following “rigidity” property justifies the terminology of the introduction.

Proposition 2.4 Let (σ_{0}, σ_{1}, σ_{∞}) be an admissible triple in SL_{2}(E) with σ_{1} unipotent. Then there exists a rigid representation

ϱ : G_{K_{n}(t)} ⟶ GL_{2}(E)

whose monodromy matrix at t = j is equal to σ_{j}. Furthermore, if ϱ′ is any irreducible rigid representation whose monodromy matrices are conjugate to those of ϱ, then ϱ′ is conjugate to ϱ ⊗ χ, where χ : G_{K_{n}} ⟶ E^{×} is a constant central character.

This follows from theorems 1 and 2 of [Be]. (See also the discussion in section 1 of [Da2].)

3 Hypergeometric abelian varieties

For the following definition, let K be any real abelian field, and O_{K} its ring of integers. (We will also write O_{n} := ℤ[ζ_{n} + ζ_{n}^{−1}] to denote the ring of integers of K_{n}.)

Definition 3.1 A hypergeometric abelian variety with multiplications by K is an abelian scheme A over (ℙ^{1} −{0, 1, ∞})_{/ℚ} of dimension [K : ℚ] equipped with an inclusion

ι : O_{K} ↪ End_{K(t)}(A)

which is compatible with the natural action of Gal(K/ℚ) on both sides, and whose associated monodromy representation is irreducible.

Define an admissible triple (σ_{0}, σ_{1}, σ_{∞}) in SL_{2}(O_{K}) in the obvious way (replacing E by O_{K} in definition 2.3). Given a hypergeometric abelian variety A with multiplications by K, one can associate to it an admissible triple (σ_{0}, σ_{1}, σ_{∞}) in SL_{2}(O_{K}) by letting σ_{j} be the image of γ_{j} acting on the DeRham cohomology H_{Dr}^{1}(A) (viewed as a two-dimensional K-vector space). Conversely, given an admissible triple (σ_{0}, σ_{1}, σ_{∞}) in SL_{2}(O_{K}), let n_{j} be the order of the semisimplification of σ_{j} and set n = lcm(n_{0}, n_{1}, n_{∞}). One sees that K must contain the fields K_{n_{j}} generated by the traces of the σ_{j}. Assume that K = K_{n}.

Proposition 3.2 Assume that σ_{1} is unipotent. There exists a hypergeometric abelian variety A with multiplications by K_{n} whose associated monodromies are (σ_{0}, σ_{1}, σ_{∞}). The isogeny class of this abelian variety depends only on the triple (σ_{0}, σ_{1}, σ_{∞}) (modulo conjugation by GL_{2}(E)).

Proof: See [Ka], sec. 5.4, or [CW], sec. 3.3. The hypergeometric abelian varieties are constructed as appropriate quotients of the Jacobians of the curves

y^{n} = x^{a}(x − 1)^{b}(x − t)^{c}.

From hypergeometric abelian varieties to rigid representations: If A is a hypergeometric abelian variety with multiplications by K, the ℓ-adic Tate module

T_{ℓ}(A) := lim_{←} A[ℓ^{k}]

is a free module of rank two over O_{K} ⊗ ℤ_{ℓ}. The natural action of G_{ℚ(t)} on this Tate module is semilinear, in the sense that

α(s · v) = s^{α} · α(v), for α ∈ G_{ℚ(t)}, s ∈ O_{K} ⊗ ℤ_{ℓ}, v ∈ T_{ℓ}(A).

In particular, if φ is a homomorphism from O_{K} to E, then T_{ℓ}(A) ⊗_{φ} E is a two-dimensional E-vector space on which G_{K(t)} acts linearly. It gives rise to a rigid two-dimensional Galois representation ϱ of G_{K(t)}, and thus to a family of representations ϱ[x] of G_{K} for all x ∈ K − {0, 1}.

Definition 3.3 The hypergeometric abelian variety A is said to be modular at x if ϱ[x] is associated to a Hilbert modular form over K with coefficients in E, for all choices of (φ, E). We say that A is modular if it is modular at x, for all x ∈ ℚ.

Remark: The representations ϱ[x] attached to A, as ℓ, E, and φ vary, form a compatible system of ℓ-adic representations of G_{K}, and hence to prove that A is modular at x, it suffices to prove that ϱ[x] is modular for a single E ⊂ ℚ̄_{ℓ}.

Examples:

1. If σ_{0}, σ_{1}, and σ_{∞} ∈ SL_{2}(ℤ) are quasi-unipotent with eigenvalues 1, 1, −1, then A is isogenous to the Legendre family of elliptic curves

y^{2} = x(x − 1)(x − t).

The modularity of A is thus a special case of the Shimura-Taniyama conjecture which was completely established by Wiles [W3].

2. If σ_{0} and σ_{∞} are of order 4 and 3 respectively, and σ_{1} is unipotent, then A_{/ℚ(t)} is isogenous to (a twist of) the universal family of elliptic curves of invariant j = 1728/(t − 1). The modularity of A in this case is merely a re-formulation of the Shimura-Taniyama conjecture.

3. If σ_{0} and σ_{1} are unipotent and σ_{∞} is of order r with r an odd prime, the corresponding hypergeometric abelian variety is the Jacobian of the hyperelliptic curve with real multiplications by ℚ(ζ_{r})^{+} given by the equation

y^{2} = (x + 2)(f(x) + 2 − 4t),

where f(x) = xg(x^{2} − 2) and g(x) is the characteristic polynomial of −(ζ_{r} + ζ_{r}^{−1}). This curve had already been considered in [TTV], and used in [Da2] to study the generalized Fermat equation x^{p}+y^{p} = z^{r}. In the language of [Da2], the mod p representations attached to A are the “even Frey representations” associated to the generalized Fermat equation x^{p} + y^{p} = z^{r}.

4. If σ_{0} and σ_{1} are unipotent and σ_{∞} has order 2r with r an odd prime, then A is the Jacobian of the hyperelliptic curve (also used in the study of x^{p} + y^{p} = z^{r})

y^{2} = f(x) + 2 − 4t.

From rigid representations to hypergeometric abelian varieties:

Let ϱ be a rigid representation of G_{K_{n}}(t) with unipotent monodromy at t = 1, associated to an admissible triple (σ_{0}, σ_{1}, σ_{∞}) in SL_{2}(E). This triple can be lifted to an admissible triple (σ̃_{0}, σ̃_{1}, σ̃_{∞}) in SL_{2}(O_{n}), i.e., there is a homomorphism φ : O_{n} ⟶ E such that φ(σ̃_{j}) = σ_{j}, and σ̃_{1} is unipotent. Let A be the hypergeometric abelian variety with multiplications by K_{n} associated to (σ̃_{0}, σ̃_{1}, σ̃_{∞}) by proposition 3.2. Then we have:

Proposition 3.4 The representation ϱ is equivalent to (a twist of) the Galois representation obtained from the action of G_{K(t)} on T_{ℓ}(A) ⊗_{φ} E.

Proof: This is a direct consequence of the uniqueness statement of proposition 2.4, since the representation associated to T_{ℓ}(A)⊗_{φ}E is a rigid representation associated to the triple (σ_{0}, σ_{1}, σ_{∞}).

Thanks to proposition 3.4, it is enough to show that all hypergeometric abelian varieties with unipotent monodromy at t = 1 are modular in order to prove theorem 1.2.

4 Congruences

Let A be a hypergeometric abelian variety with multiplication by K = K_{n}, and let (σ_{0}, σ_{1}, σ_{∞}) be the associated admissible triple in SL_{2}(O_{K}). Assume that σ_{1} is unipotent, and let ℓ be an odd prime which divides n = lcm(n_{0}, n_{1}, n_{∞}). For j = 0, 1, ∞, let n′_{j} be the prime-to-ℓ part of n_{j}, let n′ be the prime-to-ℓ part of n, and let K′ = ℚ(ζ_{n′})^{+}. Choose a prime λ of K above ℓ, and let λ′ be the unique prime of K′ below it. The prime λ′ is totally ramified in K/K′, so that the residue fields of K and K′ at λ and λ′ respectively are canonically isomorphic. Let 𝔽 be this common residue field. It is equipped with maps φ : O_{K} ⟶ 𝔽 and φ′ : O_{K′} ⟶ 𝔽. Let (σ′_{0}, σ′_{1}, σ′_{∞}) be a lift of (φ(σ_{0}), φ(σ_{1}), φ(σ_{∞})) to an admissible triple in SL_{2}(O_{K′}), and let A′ be the abelian variety associated to it by proposition 3.2.

Because G_{ℚ(t)} acts semi-linearly on A[ℓ] ⊗_{φ} 𝔽 and because λ is totally ramified in K/K′, the action of G_{K(t)} on this 𝔽-vector space extends to a linear action of G_{K′(t)}.

Theorem 4.1 The G_{K′(t)} representation A[ℓ] ⊗_{φ} 𝔽 is isomorphic to (a twist of) the representation A′[ℓ] ⊗_{φ′} 𝔽.

Proof: A direct consequence of proposition 3.4.

5 Proof of the main result

Theorem 5.1 Let A be a hypergeometric abelian variety with multiplications by K_{n}, and let (σ_{0}, σ_{1}, σ_{∞}) be the associated admissible triple. Assume that σ_{1} is unipotent, and that 8 does not divide n. If conjecture 1.1 is true, then A is modular.

Proof: The proof is by induction on d = [K_{n} : ℚ]. If d = 1, then A is an elliptic curve over ℚ(t) and the modularity of A follows from the Shimura-Taniyama conjecture, which itself follows from conjecture 1.1. If d > 1, then n is divisible by an odd prime ℓ, by the assumption that 8 does not divide n. Adopting the notation of section 4, we begin by showing (for a fixed t = x ∈ ℚ) that A[ℓ] ⊗_{φ} 𝔽 is associated to a Hilbert modular form f_{ℓ} over K. If A[ℓ] ⊗_{φ} 𝔽 is a reducible representation of G_{K}, then one may express f_{ℓ} in terms of Eisenstein series. Assume that A[ℓ] ⊗_{φ} 𝔽 is irreducible. Since n′ < n and d′ = [K′ : ℚ] < d, the induction hypothesis implies that A′ is modular. Hence so is the rigid representation A′[ℓ] ⊗_{φ′} 𝔽; let f′_{ℓ} be the associated Hilbert modular form mod ℓ on GL_{2}(K′). By theorem 4.1, the G_{K′} module A[ℓ] ⊗_{φ} 𝔽 is isomorphic to A′[ℓ] ⊗_{φ′} 𝔽, and so corresponds to the same f′_{ℓ}. Letting f_{ℓ} be the cyclic base change lift (from K′ to K) of f′_{ℓ}, it follows that the representation A[ℓ]⊗_{φ} 𝔽 is modular over K. The λ-adic Tate module T_{ℓ}(A) ⊗ K_{λ} is a potentially semistable Galois representation, since it arises from the torsion points of an abelian variety. Hence it is modular, by conjecture 1.1.

Remark: The proof that A is modular involves repeated applications of the lifting conjecture 1.1, once with each odd prime ℓ dividing n. In light of the results in [SW], it might be feasible to prove unconditionally the modularity of A at t = x , when x is such that A is ordinary at all these primes. There are infinitely many values of x with this property: for example, all the x for which n divides the numerator of x − 1.

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