On Ordinary forms and Ordinary Galois representations

Kirti Joshi & Chandrashekhar Khare School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Bombay 400 005, INDIA

e-mail: orion@tifrvax.tifr.res.in, kirti@motive.math.tifr.res.in, shekhar@motive.math.tifr.res.in

May 17, 1995

1 Introduction

Let p ≥ 5 be a prime. A well-known conjecture of Serre (see [15]) asserts that any two dimensional irreducible representation

ρ : Gal(Q̄/Q) → GL_{2}(F̄_{p}),

which is odd (in the sense that det(c) = −1, where c is complex conjugation), arises from reduction modulo p of the p-adic representation attached to a cusp form by Deligne (see [4]). In such a situation, we say that ρ is a modular mod p Galois representation. To make this more precise, let us fix once for all an embedding ι_{p} : Q̄ ↪ Q̄_{p}. This at once provides us with a place of Q̄ lying over p; we will denote this place by ℘. In what follows we will practice the following abuse of terminology. We will often speak of “congruences modulo p” to mean that we have a congruence with respect to ℘ provided by ι_{p}. Serre’s conjecture, which we have alluded to above, predicts that there is a cusp form f in the space S_{k}(Γ_{0}(N), χ) of cusp forms of weight k, level N and Nebentype χ, such that for any prime ℓ not dividing pN, we have

Tr(ρ(Frob_{ℓ})) = a_{ℓ} mod ℘

and det(ρ(Frob_{ℓ})) = χ(ℓ)ℓ^{k−1} mod ℘, where a_{ℓ} is the eigenvalue of the Hecke operator T_{ℓ}, for ℓ ≠ p. In fact, Serre also predicted that there exists certain minimal invariants N(ρ), k(ρ), ε(ρ), where ε(ρ) is a primitive character of conductor dividing N(ρ), and a form f of weight k(ρ), level N(ρ) and nebentype ε(ρ), which gives rise to the representation ρ. We will call these the Serre invariants of ρ. We note that N(ρ) is prime to p, 2 ≤ k(ρ) ≤ p^{2} − 1 and ε(ρ) is a character of conductor dividing N(ρ) and order prime to p; we refer the reader to [15] for the precise definitions of these invariants. As a consequence of the work of Mazur, Ribet, Carayol, Edixhoven and Diamond (see [5]) one now knows that if ρ is modular, then it does indeed arise from a cusp form with Serre invariants.

In general several different modular forms can give rise to the same mod p Galois representation. That this can be a source of a rich structure was perhaps first noted by Hida and Mazur. The question of diverse modular data associated a given modular ρ has been subsequently studied by several mathematicians, notably by Fred Diamond and Richard Taylor (see [6]).

This note addresses itself to one such question (see Question 1 below), which as far as we are aware, has not been addressed before. We are interested in studying ordinary Galois representations and the different new forms which give rise to them. We will make this precise in the next section. The main results of this note are Theorems 2.2 and 2.5.

We would like to thank Dipendra Prasad and H. Hida for conversations. The first author would also like to thank J. Tilouine for numerous conversations.

2 Mod p Ordinarity

Recall that we have fixed an embedding ι_{p} : Q̄ ↪ Q̄_{p}. This provides us with a place ℘ lying over p in Q̄. Let D_{p} be the decomposition group of p, with respect to the place ℘, in Gal(Q̄/Q); let I_{p} be the inertia subgroup of D_{p}. If f is a cusp form, we will frequently use the symbol π(f) to denote the corresponding automorphic representation and π_{p}(f) for its p-component.

We begin by recalling:

Definition: Let ρ : Gal(Q̄/Q) → GL_{2}(F̄_{p}) be an irreducible Galois representation. We say that ρ is ordinary at p if ρ|_{D_{p}} is reducible and has an unramified one-dimensional quotient.

Definition: A newform of level N, p not dividing N (resp. p dividing N) on Γ_{1}(N) is said to be ordinary at p if the eigenvalue of T_{p} is a p-adic unit (resp. the eigen value of U_{p} on f is a p-adic unit).

Remark: If f ∈ S_{k}(Γ_{1}(N)), with N prime to p, is ordinary at p, then from f we can create in the usual way an ordinary p-old form f′ ∈ S_{k}(Γ_{1}(Np)) by the usual recipe. We recall the recipe. If α, β are the roots of the Euler factor at p of L(f, s), then we may assume, without loss of generality, that α is a ℘-unit. Then f′(z) = f(z) − βf(pz) is the required form.

The following result of Deligne, Hida and Mazur-Wiles, which is wellknown, is the starting point for our question.

Theorem 2.1 Let f ∈ S_{k}(Γ_{0}(N)) be an Hecke eigen form which is a new form. Assume that the eigen value of T_{p} (or U_{p}) is a p-adic unit. Also assume that the mod p representation ρ_{f,p} is irreducible. Then ρ_{f,p}|_{D_{p}} is ordinary at p.

This motivates the following question.

Question 1 Suppose ρ is an irreducible representation. Assume that ρ is ordinary and is modular. Then is there at least one ordinary modular form which gives rise to ρ?

The following theorem provides a reasonable answer to this question.

Theorem 2.2 Assume that we are given a modular mod p Galois representation ρ as above.

1. If ρ is ordinary at p then there is one (and hence infinitely many) ordinary newform(s) which give rise to ρ.

2. If ρ is not ordinary, then no ordinary form can give rise to it.

3. Given a ρ as above, there are infinitely many distinct newforms which are non-ordinary at p which give rise to ρ. Moreover these may be chosen to be either principal series or supercuspidal at p.

Remark: If f gives rise to ρ then one can twist f by a character of p-power conductor and p-power order to get non-ordinary forms which give rise to ρ. But using the theorem stated in the introduction of [11], we can get nonordinary forms which give rise to ρ in other less obvious ways.

Proof: We assume that ρ is modular and ordinary at p, i.e

ρ|_{I_{p}} ∼ ( \begin{matrix}χ & ∗ \\ 0 & 1 \end{matrix} ) ,

where χ is a power of the mod p cyclotomic character ω (recall that ω : Gal(Q̄_{p}/Q_{p}) → F_{p}^{∗} is defined by σ(ζ_{p}) = ζ_{p}^{ω(σ)}, where ζ_{p} is a primitive pth root of unity). Thus χ = ω^{a} for some integer a. We normalise a to be between 1 and p − 1. Then Serre defines:

k(ρ) = \{ \begin{matrix}1 + a & if a ≠ 1 \\ 2 & if a = 1 and ρ is peu ramifiée at p \\ p + 1 & if a = 1 and ρ is très ramifiée at p \end{matrix}

As a consequence of Edixhoven’s work (see [7]), it is known that ρ arises from a form of weight k(ρ) on Γ_{1}(N) for some N such that (N, p) = 1. From the above we see that 2 ≤ k(ρ) ≤ p + 1. We now need to recall theorems of Deligne and Fontaine (see Theorem 2.5 and 2.6 of [7] respectively).

Theorem 2.3 (Deligne) Let f be a newform of level N prime to p and weight k, with 2 ≤ k ≤ p + 1. If a_{p} ≢ 0 mod ℘ then ρ|_{D_{p}} is reducible and has an unramified quotient of dimension one.

Theorem 2.4 (Fontaine) With the same hypothesis as the above theorem, if a_{p} ≡ 0 mod ℘, then ρ|_{D_{p}} is irreducible.

Thus it follows from these two theorems that any form f ∈ S_{k(ρ)}(Γ_{1}(N)) which gives rise to ρ is ordinary. After this the existence of infinitely many ordinary newforms which give rise to ρ follows from the work of Hida (see [10]).

For the second assertion, we just need remark that this is a result of Deligne, Hida and Mazur-Wiles quoted above.

For the third assertion, we just note that ρ arises from a cusp form in S_{k(ρ)}(Γ_{1}(N(ρ)). Then by the main theorem of [11], we see that ρ arises from a cusp form in S_{2}(N(ρ)p^{r}, ε(ρ))^{p−new} for all r ≥ 3. Any such form is supercuspidal at p if r is odd (see Remark 4.25 of [8]). If r is even then the form in the p-new part of S_{2}(N(ρ)p^{r}, ε(ρ)) which gives rise to ρ can be chosen to be principal series at p. This is an application of Carayol’s lemma (see [11], Remarks 11.2 and 11.3). From this the third assertion of the theorem follows from the result that a form in the p-new part of S_{2}(Γ_{0}(N(ρ)p^{r}), χ) with cond(χ) = p^{s} and s < r (r > 1) has p-eigenvalue zero (see [13]). □

Remark: It is evident after the theorem that there exists a Hida family which gives rise to the representation ρ. We do not know if the non-ordinary forms which give rise to ρ occur in p-adic families. This may follow from Coleman’s work on the conjectures on slopes of Hecke operators of Gouvêa and Mazur (see [9]).

This finishes the proof of the theorem. We now consider another aspect of Question 1. We have seen in the above theorem that if we start with a mod p modular representation ρ which is ordinary, then any form of minimal weight and level prime to p which gives rise to ρ is ordinary. In the theorem which follows we prove a similar result about the forms of weight two and minimal p-power level which give rise to ρ.

Theorem 2.5 Let ρ be an irreducible modular mod p representation which is ordinary at p. Then ρ arises from a form f ∈ S_{2}(Γ_{1}(N(ρ)p)) with N(ρ) prime to p. Further any form in S_{2}(Γ_{1}(Np)) (for any N which satisfies (N, p) = 1) which gives rise to ρ is ordinary at p if k(ρ) ≠ 2; otherwise the form in S_{2}(Γ_{1}(N)) is ordinary while there exists a p-old form in S_{2}(Γ_{1}(Np)) which is not ordinary and which gives rise to ρ.

Proof: That ρ arises from S_{2}(Γ_{1}(N(ρ)p)) follows from the level lowering results of Mazur, Ribet, Diamond and Carayol and Edixhoven’s proof of the weight part of Serre’s conjecture upon using the result (see [1]) that a representation ρ which arises from S_{k}(Γ_{1}(N(ρ))) (2 ≤ k ≤ p + 1) also arises from S_{2}(Γ_{1}(N(ρ)p)). To prove the second part of the assertion we split the argument into two cases.

Suppose first that ρ|_{I_{p}} ∼ ( \begin{matrix}ω & ∗ \\ 0 & 1 \end{matrix} ), where ω is the mod p cyclotomic

character. If ρ is finite at p then we know that ρ arises from a form f in S_{2}(Γ_{1}(N)) (for some N prime to p, this is a consequence of Mazur’s principle cf. [14]) else ρ arises from a form in S_{2}(Γ_{1}(N) ∩ Γ_{0}(p))^{p−new}. In the first sub case we are done by another application of theorems of Deligne and Fontaine quoted above. If ρ arises from S_{2}(Γ_{1}(N) ∩ Γ_{0}(p))^{p−new} (with (N, p) = 1) then we use the well-known result that a form in this space has p-th eigen value such that its square is ε(p), where ε is the nebentype and is consequently ordinary at p.

Now suppose that ρ|_{I_{p}} ∼ ( \begin{matrix}ω^{a} & ∗ \\ 0 & 1 \end{matrix} ) with a ≢ 1 mod p − 1. Assume that

a non-ordinary p-newform f in S_{2}(Γ_{1}(Np)) gives rise to ρ. In this case we use Edixhoven’s proof of Fontaine’s theorem (Theorem 2.6 of [7]). We follow Edixhoven’s notations. Let J_{1}(Np)_{/Z[ζ_{p}]^{+}} be the Neron model of the jacobian of X_{1}(Np)_{/Q(ζ)^{+}} where Q(ζ_{p})^{+} is the totally real subfield of Q(ζ_{p}); Z[ζ_{p}]^{+} the ring of integers of Q(ζ_{p})^{+}. Let J_{1}(Np)_{/F_{p}} be the special fibre of the Neron model, and let J_{1}(Np)_{/F_{p}}^{0} be its connected component. Let I, E be the Igusa and the exotic Igusa curves of level Np; J_{I}, J_{E} their jacobians. We follow Edixhoven closely here. Then J_{I} × J_{E} is the maximal abelian variety quotient of J_{1}(Np)_{/F_{p}}^{0}, and the kernel is a torus T. Thus we have an exact sequence of commutative groups schemes (in the flat topology):

0 → T → J_{1}(Np)_{/F_{p}}^{0} → J_{I} × J_{E} → 0.

This gives rise to the following exact sequence of p-divisible groups over F_{p}

0 → T [p^{∞}] → J_{1}(Np)_{/F_{p}}^{0}[p^{∞}] → J_{I}[p^{∞}] ⊕ J_{E}[p^{∞}] → 0.

Let m be the maximal ideal of the Hecke algebra corresponding to ρ. Then we have an induced decomposition (of finite group schemes over F_{p}):

J_{1}(Np)[m] = J_{I}[m] ⊕ J_{E}[m]

where we have used the fact that the diamond operator action of (Z/p)^{∗} on T is trivial (see the proof of Theorem 2.6 in [7]); on the other hand J_{1}(Np)[m] has non trivial action of this group as f has non trivial Nebentype at p. Thus ρ lives in J_{I}[m] ⊕ J_{E}[m]. Then using the formula for relating the Frobenius F_{p}, Verschiebung V_{p} and U_{p} in Theorem 5.3 of Wiles (see [17]) we have the following relations on J_{I}: U_{p} = V_{p} < n_{p} > (where n_{p} is a suitable integer) and on J_{E} we have U_{p} = F_{p}+∑_{ζ≠1} w_{ζ}. Then as in the proof of Theorem 5.4 of [17], we see that ∑_{ζ≠1} w_{ζ} = 0. By assumption, f is not ordinary so that eigen value of U_{p} is zero. Thus we see that V_{p} (resp. F_{p}) operates trivially on J_{I} (resp. on J_{E}). Now we use the well-known decomposition of a finite group scheme over F_{p} into its étale, multiplicative, local-local pieces. Writing G_{I}, G_{E} for these two components in the above direct sum, we further write G_{I}^{e}, G_{I}^{m}, G_{I}^{l} for the étale, multiplicative and local-local components respectively (and similarly for G_{E}). We see from the above action of F_{p} and V_{p} that G_{I}^{m} = G_{E}^{e} = 0. Then as Edixhoven argues, G_{I}, G_{E} are Cartier duals over F_{p}(ζ_{N}); moreover as is shown on page 583 of [7], this is a perfect pairing. Using this pairing, Edixhoven shows that G_{E} is local-local, i.e., F_{p} and V_{p} are both nilpotent on G_{E}. Since the category of local-local group schemes is closed under taking Cartier duals, we see that G_{I} is also a local-local group scheme. Thus G_{I} and G_{E} are both local-local group schemes. This contradicts ordinarity of ρ. □

Remark: This result is best possible as there are forms in S_{2}(Γ_{0}(Np^{2}))^{p−new} (which can be even supercuspidal at p: see Remark 11.1 of [11]) which give rise to ordinary mod p representations though any form in this space has p^{th} eigenvalue zero. This may be seen from Theorem 3 of [11], where it is shown that if a mod p representation arises from S_{2}(Γ_{0}(N)), then it also arises from the p-new part of S_{2}(Γ_{0}(Np^{2})).

We take this opportunity to explicate a question which is implicit in [11]. Let f ∈ S_{2}(Γ_{0}(N)), (N, p) = 1 be a newform giving rise to an irreducible

Galois representation ρ defined over F̄_{p}. Then in [11] (see the theorem in the introduction) it is shown that ρ also arises from forms in the p-new part of S_{2}(Γ_{0}(Np^{r}), ε) (with r ≥ 3) where ε is any character with conductor dividing p^{r} and of p-power order. If r = 2n + 1 and cond(ε)|p^{n} these forms are supercuspidal at p. For r ≡ 0 mod 2, using Carayol’s lemma (see [11], Remarks 11.2 and 11.3), it follows the the forms in the p-new part of S_{2}(Γ_{0}(Np^{r})) may be chosen to be principal series at p. Thus we may ask the question:

Question 2 Can one always find a form f ∈ S_{2}(Γ_{0}(Np^{2n}), ε)^{p−new} with cond(ε) = p^{k} (k < n) and of p-power order, which gives rise to ρ and such that it is supercuspidal at p?

Remark: After this note was written, the above question has been answered affirmatively by the second author (see [12]).

3 p-adic Ordinarity

Now we turn to the p-adic analogue of the above question. To fix notations, we work with a p-adic ring O_{p}, which we will assume to be large enough so that the eigen values of Hecke operators of the form in question lie in it. Typically, we can take this to be the ring of integers in the completion of the coefficient field E_{f} of f along the valuation given by ℘. We now recall the definition of ordinarity.

Definition: Let σ : Gal(Q̄/Q) → GL_{2}(O_{p}) be an absolutely irreducible Galois representation. We say that it is ordinary at p if σ|_{D_{p}} is reducible and has an unramified one-dimensional quotient.

We are then concerned with the following question, which was also raised by Deligne in a letter to Serre (see [3]).

Question 3 Let σ : Gal(Q̄/Q) → GL_{2}(O_{p}) be the p-adic representation attached to a newform such that the residual representation is absolutely irreducible and such that σ is ordinary at p. Then is the corresponding form ordinary?

We also have a related question.

Question 4 Let f be a newform such that the corresponding automorphic representation π_{p}(f) is supercuspidal and let σ be the p-adic representation associated to f. Then is σ|_{D_{p}} irreducible?

The first question may have an affirmative answer as consequence of the recent work of Wiles (see [16]), which establishes at least in the semistable case, the identity of a certain universal ring of ordinary Galois deformations with the ordinary Hecke algebra of Hida. We hope to come back to this in a future work.

Remark: As Dipendra Prasad has remarked to us, it can be easily seen that the second question has an affirmative answer if f is a CM form.

Remark: From the theorem in the introduction of [11] we see that if ρ arises from a form f in S_{2}(Γ_{0}(N)) with (N, p) = 1 (ρ absolutely irreducible), then ρ also arises from the p-new part of S_{2}(Γ_{0}(Np^{2m+1})) (with m ≥ 1). As noted above any such newform is supercuspidal at p. This illustrates that the mod p analogue of the above question is false (as there are many such f’s which give rise to irreducible ρ, and which are ordinary at p).

Remark: The main reason why the above question seems difficult is because using Langlands-Carayol (see [2]), one cannot obtain information about the p-adic representation associated to a newform restricted to D_{p}, from information about π_{p}(f). Observe that if π_{p}(f) is supercuspidal then the ℓ-adic (ℓ ≠ p) representation of f restricted to D_{p} is irreducible.

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