CHAPTER 6

Appendix by Kevin Buzzard: A mod ℓ multiplicity one result

In this appendix, we explain how the ideas of [45] can be used to prove the following mild strengthening of the multiplicity one results in §9 of [31].

The setup is as follows. Let f be a normalised cuspidal eigenform of level N, and weight k, defined over \overline{F}_{ℓ}, with ℓ ∤ N and 2 ≤ k ≤ ℓ + 1. Let N^{∗} denote N if k = 2, and Nℓ if k > 2. Let J_{Q} be the Jacobian of the curve X_{1}(N^{∗})_{Q}, and let H denote the Hecke algebra in End(J_{Q}) generated over Z by T_{p} for all primes p, and all the Diamond operators of level N^{∗}. It is well-known (for example by Proposition 9.3 of [45]) that there is a characteristic 0 normalised eigenform F in S_{2}(Γ_{1}(N^{∗})) lifting f. Let 𝔪 denote the maximal ideal of H associated to F (note that 𝔪 depends only on f and not on the choice of F), and let F = H/𝔪, which embeds naturally into \overline{F}_{ℓ}. Suppose the representation ρ_{f} : G_{Q} → GL_{2}(\overline{F}_{ℓ}) associated to f is absolutely irreducible, and furthermore assume that if k = ℓ + 1 then ρ_{f} is not isomorphic to a representation coming from a form of weight 2 and level N.

Theorem 6.1. If ρ_{f} is ramified at ℓ, or if ρ_{f} is unramified at ℓ but ρ_{f}(Frob_{ℓ}) is not a scalar matrix, then J_{Q}(\overline{Q})[𝔪] has H/𝔪-dimension two, and hence is a model for (precisely one copy of) ρ_{f}.

The motivation for putting ourselves in the setup above is that every absolutely irreducible modular mod ℓ representation has a twist coming from a modular form of level prime to ℓ and weight at most ℓ + 1. In particular, every modular mod ℓ representation has a twist coming from a form satisfying the conditions of our setup. Furthermore, if f is as in our setup, then Theorems 2.5 and 2.6 of [31] tell us the precise structure of the restriction of ρ_{f} to D_{ℓ}, a decomposition group at ℓ. These results are explained in Section 2.2. Using them, it is easy to deduce

Corollary 6.2. Let ρ be an absolutely irreducible modular mod ℓ representation, such that ρ_{f}(D_{ℓ}) is not contained within the scalars. Then some twist of ρ comes from a modular form satisfying the conditions of the theorem, and hence ρ is a multiplicity one representation in the sense of Remark 3.4.2.

The theorem, commonly referred to as a “multiplicity one theorem”, is a mild extension of results of Mazur, Ribet, Gross and Edixhoven. It was announced for ℓ = 2 as Proposition 2.4 of [9] but the proof given there is not quite complete—in fact, the last line of the proof there is a little optimistic. I would hence like to thank Ribet and Stein for the opportunity to correct this oversight in [9].

82 BUZZARD, A MOD ℓ MULTIPLICITY ONE RESULT

Proof of Theorem. Firstly we observe that the only case not dealt with by Theorem 9.2 of [31] is the case when k = ℓ and ρ_{f} is unramified at ℓ, with ρ_{f}(Frob_{ℓ}) a non-scalar matrix whose eigenvalues are equal. Moreover, using Theorems 2.5 and 2.6 of [31] we see that in this case f must be ordinary at ℓ. We are hence in a position to use the detailed construction of ρ_{f} given in §§11–12 of [45]. We will follow the conventions set up in the present paper for normalisations of Hecke operators, and so in particular the formulae below differ from the ones in [45] by a twist.

The maximal ideal 𝔪 of H associated to f gives rise as in (12.5) of [45] to an idempotent e ∈ H_{ℓ} := H ⊗_{Z} Z_{l}, such that the completion H_{𝔪} of H at 𝔪 is just eH_{ℓ}. Let G denote e(J_{Q_{ℓ}}[ℓ^{∞}]), the part of the ℓ-divisible group of J which is associated to 𝔪. Then H_{𝔪} acts on G, and it is proved in Propositions 12.8 and 12.9 of [45] that there is an exact sequence of ℓ-divisible groups

0 → G^{0} → G → G^{e} → 0

over Q_{ℓ}, which is H_{𝔪}-stable. Let

0 → T^{0} → T → T^{e} → 0

be the exact sequence of Tate modules of these groups. We now explain explicitly, following [45], how the group D_{ℓ} acts on these Tate modules.

If k > 2 then there is a Hecke operator U_{ℓ} in H_{𝔪}, and we define u = U_{ℓ}. If k = 2 then there is a Hecke operator T_{ℓ} in H_{𝔪} and because we are in the ordinary case we know that T_{ℓ} is a unit in H_{𝔪}. We define u to be the unique root of the polynomial X^{2} − T_{ℓ}X + ℓ⟨ℓ⟩ in H_{𝔪} which is a unit (u exists by an appropriate analogue of Hensel’s lemma).

The calculations of Propositions 12.8 and 12.9 of [45] show that, under our conventions, the absolute Galois group D_{ℓ} of Q_{ℓ} acts on T^{e} as λ(u), where λ(x) denotes the unramified character taking Frob_{ℓ} to x. Moreover, these propositions also tell us that D_{ℓ} acts on T^{0} via the character χ_{ℓ}λ(u^{−1}⟨ℓ⟩_{N})χ^{ℓ−2}, where χ_{ℓ} is the cyclotomic character and χ is the Teichmüller character. The key point is that this character takes values in H^{×}.

The next key observation is that a standard argument on differentials, again contained in the proof of Propositions 12.8 and 12.9 of [45], shows that G^{e}[𝔪] = 𝔪^{−1}ℓT^{e}/ℓT^{e} has H_{𝔪}/𝔪-dimension 1 and that G^{0}[𝔪] has dimension d^{0} ≥ 1. (Note that the fact that G^{e}[𝔪] has dimension 1 implies, via some simple linear algebra, that the sequence 0 → G^{0}[𝔪] → G[𝔪] → G^{e}[𝔪] → 0 is exact, as asserted by Gross.) Furthermore, because we can identify G^{0}[𝔪] with 𝔪^{−1}ℓT^{0}/ℓT^{0}, we see that the action of D_{ℓ} on G^{0}[𝔪] is via a character which takes values in (H/𝔪)^{×}. In particular, D_{ℓ} acts as scalars on G^{0}[𝔪].

Let us now assume that ρ_{f} is unramified at ℓ, and that ρ_{f}(Frob_{ℓ}) is a nondiagonalisable matrix with eigenvalue α ∈ H/𝔪. Choose a model ρ for ρ_{f} defined over GL_{2}(H/𝔪). By the theorem of Boston, Lenstra and Ribet, we know that G[𝔪] is isomorphic to a direct sum of d copies of ρ, or more precisely, d copies of the restriction of ρ to D_{ℓ}. Here d is an integer satisfying 2d = d^{0} + d^{e}. Hence, if G[𝔪]^{α} denotes the subspace of G[𝔪] where Frob_{ℓ} acts as α, then the H/𝔪-dimension of G[𝔪]^{α} is at most d. On the other hand, Frob_{ℓ} acts on G[𝔪]^{0} as a scalar, and hence this scalar must be α, and so we see G[𝔪]^{0} ⊆ G[𝔪]^{α}. Hence d^{0} ≤ d = (d^{0} + 1)/2. We deduce that d^{0} ≤ 1 and hence d^{0} = d = 1 and the theorem is proved.

We remark that L. Kilford has found examples of mod 2 forms f of weight 2, such that ρ_{f} is unramified at 2 and ρ_{f}(Frob_{2}) is the identity, and where J_{Q}(\overline{Q})[𝔪] has H/𝔪-dimension 4, and so one cannot hope to extend the theorem to this case. See Remark 3.6 for more details, or [63]. A detailed analysis of what is happening in this case, at least in the analogous setting of forms of weight 2 on J_{0}(p), with p prime, has been undertaken by Emerton in [38]. In particular, Emerton proves that multiplicity one fails if and only if the analogue of the exact sequence 0 → T^{0} → T → T^{e} → 0 fails to split as a sequence of H_{𝔪}-modules.

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