Modularity of a family of elliptic curves

Fred Diamond D.P.M.M.S., 16 Mill Lane University of Cambridge Cambridge, CB2 1SB, UK

Kenneth Kramer Department of Mathematics Queens College(CUNY) Flushing, NY 11367, USA

April 12, 1995

We shall explain how the following is a corollary of results of Wiles [W]:

Theorem If a, b and c are distinct rational numbers, then the elliptic curve defined by

y^{2} = (x − a)(x − b)(x − c)

is modular.

Recall that Wiles proves that if E is a semistable elliptic curve over Q, then E is modular [W, Thm. 0.4]. He begins by considering the Galois representations ρ̄_{E,3} (respectively, ρ_{E,3}) on the 3-division points (respectively, 3-adic Tate module) of E. If ρ̄_{E,3} is irreducible, then a theorem of Langlands and Tunnell is used to show that ρ̄_{E,3} arises from a modular form. Wiles deduces that ρ_{E,3} also arises from a modular form by appealing to his results in [W, Ch. 3] and those with Taylor in [TW] to identify certain universal deformation rings as Hecke algebras. This suffices to prove that E is modular if ρ̄_{E,3} is irreducible. When ρ̄_{E,3} is reducible, Wiles gives an argument which allows one to use ρ_{E,5} instead.

In fact, Wiles’ results apply to a larger class of elliptic curves than those which are semistable [W, Thm. 0.3], and this was subsequently extended in [Di] to include all elliptic curves with semistable reduction at 3 and 5. Rubin and Silverberg noted that an elliptic curve as in the above theorem necessarily has a twist which is semistable outside 2, and hence is modular by [Di, Thm.

The authors are grateful to Kevin Buzzard, Ken Ribet and Karl Rubin for comments on an earlier version of this note.

1.2]. The purpose of this note is to explain how, by a refinement of their observation, the above theorem follows directly from Wiles’ work, without appealing to [Di].

Lemma 1 (Rubin-Silverberg) By at most a quadratic twist, an elliptic curve as in the theorem may be brought to the form

E : y^{2} = x(x − A)(x + B) (1)

for some nonzero integers A and B with A and B relatively prime, B even and A ≡ −1 mod 4. Let C = A + B. If p is an odd prime, then E has either good or multiplicative reduction at p, according to whether or not p is prime to ABC.

Proof: Note that a curve as in the theorem is isomorphic to one defined by equation (1) for some integers A and B with AB(A + B) ≠ 0. Let D = gcd(A, B). Twisting by Q(\sqrt{D}), we may assume that A and B are relatively prime. By translating x or exchanging A and B, we may assume that B is even. Finally if A ≡ 1 mod 4, we twist again by Q(i).

The reduction type of E for odd primes p may be determined as in [Se2, §4] and [Si1, Ch. VII].

See [O, §I.1] for discussion of the reduction type and conductor of curves given by equation (1), but under certain restrictions in the case p = 2. See also [Da, Lemma 2.1] for a related case.

Lemma 2 Suppose that E is an elliptic curve over Q_{2} defined by the model (1), with A ≡ −1 (mod 4) and B even. The reduction type, conductor exponent f_{2}(E) and valuation of the minimal discriminant of E are given by the following table:

ord_{2}(B)  1 2 3 4 ν ≥ 5
Kodaira Symbol III I_{1}^{∗} III^{∗} I_{0} I_{2ν−8}
f_{2}(E) 5 3 3 0 1
ord_{2}(Δ_{min})  6 8 10 0 2ν − 8

Proof: A twist of E by the unramified extension Q_{2}(\sqrt{−A}) affects neither reduction type nor conductor exponent, and provides a model of the form

y^{2} = x(x + 1)(x + s) (2)

with ord_{2}(s) = ord_{2}(B) ≥ 1 and discriminant Δ = 16s^{2}(1 − s)^{2}. For the convenience of the reader, we indicate the appropriate translations of model, depending on ord_{2}(s), so that the explicit criteria of Tate’s algorithm [T] may be used.

If ord_{2}(s) = 1, then ord_{2}(Δ) = 6. Put y + x for y in (2) to get

y^{2} + 2xy = x^{3} + sx^{2} + sx. (3)

If ord_{2}(s) = 2, then ord_{2}(Δ) = 8. Put x + 2 for x in (3), to get

y^{2} + 2xy + 4y = x^{3} + (s + 6)x^{2} + (5s + 12)x + (6s + 8).

If ord_{2}(s) = 3, use the model (3) with ord_{2}(Δ) = 10. If ord_{2}(s) ≥ 4, the model (3) is not minimal and may be reduced to

y^{2} + xy = x^{3} + \frac{s}{4}x^{2} + \frac{s}{16}x (4)

with discriminant s^{2}(1 − s)^{2}/256. Thus (4) has good reduction if ord_{2}(s) = 4 and multiplicative reduction if ord_{2}(s) ≥ 5.

To show that an elliptic curve over Q is modular, we may replace it with one to which it is isomorphic over Q̄. We may therefore assume that E is defined by equation (1) with A and B as in Lemma 1. If E has good or multiplicative reduction at p = 2, then E is semistable and we can conclude from [W, Thm. 0.4] that E is modular. In view of Lemma 2, we may also assume henceforth that ord_{2}(B) = 1, 2 or 3.

Choose a basis for E[3], the kernel of multiplication by 3 on E, and let ρ = ρ̄_{E,3} denote the representation

G_{Q} → GL_{2}(F_{3})

defined by the action of G_{Q} = Gal (Q̄/Q) on E[3]. Then ρ factors through Gal (L/Q) where L = Q(E[3]) is the number field generated by the coordinates of the points of E[3].

Recall that L, and hence ρ, is unramified at p if p ≠ 3 is a prime of good reduction; i.e., if p does not divide 3ABC. For each prime p dividing 3ABC, we fix an embedding Q̄ ↪ Q̄_{p} and regard G_{p} = Gal (Q̄_{p}/Q_{p}) as a decomposition subgroup of G_{Q} at a place over p. Thus ρ|G_{p} is equivalent to the representation of G_{p} defined by its action on E[3](Q̄_{p}). Let I_{p} ⊂ G_{p} denote the inertia group.

Under our assumption that E is not semistable at p = 2, the following lemma applies to give additional information about ρ|I_{2}.

Lemma 3 If ord_{2}(B) = 1, 2 or 3, then ρ|I_{2} is absolutely irreducible.

Proof: For the moment, consider the more general case of a representation ψ : I → SL_{2}(F̄_{ℓ}), where I is the inertia group of a finite Galois extension of padic fields and ℓ ≠ p is a prime. Let b(ψ) denote the wild conductor exponent [Se2, §4.9]. If b(ψ) is odd, then ψ is irreducible. Indeed, were ψ to be reducible, it would be equivalent to a representation of the form

( \begin{matrix}χ & ∗ \\ 0 & χ^{−1} \end{matrix} ) .

But then, because b is integer-valued and additive on short exact sequences, b(ψ) = 2b(χ) would be even.

Under the hypotheses of this lemma, the elliptic curve E has additive reduction at 2 and odd conductor exponent f_{2}(E) = 2 + b(ρ|I_{2}). Since det ρ is the quadratic character associated to Q(\sqrt{−3}), the image of I_{2} under ρ is contained in SL_{2}(F_{3}). It follows that ρ|I_{2} is absolutely irreducible.

Remark The same argument shows that under the hypotheses of Lemma 3, ρ̄_{E,ℓ}|I_{2} is irreducible for all odd primes ℓ. Furthermore, as a consequence of the group structure of SL_{2}(F_{3}), the image of wild ramification is isomorphic to the quaternion group of order 8.

From the lemma, we see that ρ is irreducible and hence modular by the theorem of Langlands and Tunnell (see [W, Ch. 5]), and we need only verify that ρ satisfies certain local conditions in order to apply [W, Thm. 0.3]. Since it is left to the reader of [W] to verify that the local conditions in [W, Thm. 0.3] are sufficient to apply the central result [W, Thm. 3.3], we shall explain directly how this is done in the case with which we are concerned.

For completeness, we recall how the Tate parametrization is used for primes p at which E has multiplicative reduction. (See [Se2, §2.9].) We let F denote the unramified quadratic extension of Q_{p} in Q̄_{p}. Then E has split multiplicative reduction over F and the Tate parametrization (see [Si2, §V.3]) provides an isomorphism

Q̄_{p}^{×}/q^{Z} ≅ E(Q̄_{p})

of Gal (Q̄_{p}/F)-modules for some q ∈ Q_{p} with ord_{p}(q) > 0. From this it follows that for each prime ℓ, there is a filtration of Gal (Q̄_{p}/F)-modules

0 → Z_{ℓ}(1) → T_{ℓ}(E) → Z_{ℓ} → 0,

where T_{ℓ}(E) is the ℓ-adic Tate module and Z_{ℓ}(1) = lim_{←} μ_{ℓ^{n}}(Q̄_{p}). One then checks that the representation of G_{p} on T_{ℓ}(E) is equivalent to one of the form

χ ⊗ ( \begin{matrix}ϵ & ∗ \\ 0 & 1 \end{matrix} )

where χ is either trivial or the unramified quadratic character of G_{p} and ϵ is the cyclotomic character given by the action of G_{p} on Z_{ℓ}(1). It follows that the representation of G_{p} on E[ℓ] is of this form as well, but with ϵ now defined by the action of G_{p} on μ_{ℓ}.

Let us now return to the case ℓ = 3, and suppose first that p > 3 is a prime dividing ABC. Then the above analysis applies to ρ|G_{p} and shows that ρ is either unramified or type (A) at p in the terminology of [W, Ch. 1]. (The first possibility occurs precisely when ord_{p}(ABC) is divisible by 3; see [Se2, §4].)

Suppose next that p = 3. If 3 divides ABC, then the above analysis shows that ρ|G_{3} is ordinary at p in the terminology of [W, Ch. 1]. If on the other hand 3 does not divide ABC, then the elliptic curve E has good reduction at 3. In fact the equation (1) defines an elliptic curve E over Z_{3} such that E_{Q_{3}} is isomorphic to E_{Q_{3}} (see [Si2, §IV.5]). The kernel of multiplication by 3 on E is a finite flat group scheme E[3] over Z_{3}. The representation ρ|G_{3} is given by the action of G_{3} on E[3](Q̄_{3}), which we may identify with E[3](Q̄_{3}). In this sense, ρ|G_{3} arises from a finite flat group scheme over Z_{3}. Now ρ|G_{3} is reducible if and only if E has ordinary reduction at 3; i.e., if and only if E_{F_{3}} is ordinary. In that case ρ is ordinary at 3 in the sense of [W]. On the other hand, ρ|G_{3} is irreducible if and only if E_{F_{3}} is supersingular, in which case ρ is flat at 3 in the sense of [W, Ch. 1].

Finally, suppose that p = 2, so that ρ|I_{2} is absolutely irreducible by Lemma 3. We claim that ρ|G_{2} is type (C) at 2 in the terminology of Wiles [W, Ch. 1]. Recall that this means that H^{1}(G_{2}, W) = 0, where W is the G_{2}-module of endomorphisms of E[3](Q̄_{2}) of trace zero. From the triviality of the local Euler characteristic ([Se1, Thm. II.5]), we have

#H^{1}(G_{2}, W) = #H^{0}(G_{2}, W) · #H^{2}(G_{2}, W).

By local Tate duality ([Se1, Thm. II.1]), we have

#H^{2}(G_{2}, W) = #H^{0}(G_{2}, W^{∗})

where W^{∗} = Hom(W, μ_{3}). Therefore we wish to prove that H^{0}(G_{2}, W) and H^{0}(G_{2}, W^{∗}) both vanish. But in fact H^{0}(I_{2}, W) and H^{0}(I_{2}, W^{∗}) already vanish. Indeed, I_{2} acts trivially on μ_{3}, from which we deduce that there is a (noncanonical) isomorphism W^{∗} ≅ W of I_{2}-modules; hence it suffices to show that H^{0}(I_{2}, W) = 0. Since I_{2} acts absolutely irreducibly on W, Schur’s lemma implies that the only I_{2}-invariant endomorphisms of F̄_{3}^{2} are scalars. But the only scalar in W is zero.

We may now conclude that the representation ρ_{E,3} of G_{Q} on T_{3}(E) arises from a modular form. Indeed Wiles [W, Thm. 3.3] establishes an isomorphism between the universal deformation ring of type D and the Hecke algebra T_{D}, where D = (·, Σ, Z_{3}, ∅) with

• · as flat or Selmer according to whether or not E has supersingular reduction at 3;

• Σ as the set of primes dividing 3ABC.

Since ρ_{E,3} defines a deformation of ρ of type D, the universal property of the deformation ring thus provides a homomorphism T_{D} → Z_{3} with the following property. For all p not dividing 3ABC, the Hecke operator T_{p} is sent to a_{p} = p + 1 − N_{p} where N_{p} is the number of F_{p}-points on the reduction of E mod p. The definition of T_{D} ensures that this homomorphism arises from a normalized eigenform of weight two whose p^{th} Fourier coefficient is a_{p} for all such p. Hence E is modular.

References

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