Local splitting of ordinary

Galois representations

Eknath Ghate Hyderabad, 2003.

This is joint work with V. Vatsal.

Cusp forms

Let f = ∑_{n=1}^{∞} a(n, f) q^{n} be a cusp form of weight k ≥ 2, level N ≥ 1, character χ : (ℤ/N)^{×} → ℂ^{×}. f(\frac{az+b}{cz+d} ) = χ(d)(cz+d)^{k}f(z) for all ({a \atop c} {b \atop d}) ∈ Γ_{0}(N). Assume f is a primitive form: f is a newform and a normalized eigenform. Let p be a prime number. Let ℘ be a prime of (every subfield of) \overline{ℚ} lying over p. Let K_{f} = ℚ(a(n, f)) be the number field generated by the Fourier coefficients of f.

Ordinary Galois representations There is a Galois representation (Deligne)

ρ_{f} = ρ_{f,℘} : Gal(\overline{ℚ}/ℚ) → GL_{2}(K_{f,℘}) such that for all primes ℓ outside Np

• trace ρ_{f}(Frob _{ℓ}) = a(ℓ, f),

• det ρ_{f}(Frob _{ℓ}) = χ(ℓ)ℓ^{k−1}. Def.f is ordinary at ℘ if a(p, f) is a ℘-adic unit. Let G_{p} be the decomposition group at ℘|p. Theorem. (Mazur-Wiles, Wiles) f ordinary at ℘ ⟹ ρ_{f}|_{G_{p}} is ‘upper-triangular’:

ρ_{f}|_{G_{p}} ∼ (\begin{matrix}δ & u \\ 0 & ϵ \end{matrix})

where ϵ, δ : G_{p} → K_{f,℘}^{×}, with ϵ unramified and u : G_{p} → K_{f,℘}. Eg. If f ↔ E with gd. ord. reduction at p, then on I_{p}: 0 → ℤ_{p}(1) → T_{p}(E) → T_{p}(Ẽ) = ℤ_{p} → 0.

Greenberg’s question

Def. ρ_{f} is split at p if ρ_{f}|_{G_{p}} is semi-simple ( ⟺ u = 0 in some basis). Question. (Greenberg) When is ρ_{f} split at p? Def. f has CM if there is a character θ such that a_{p} = a_{p} · θ(p) for all but finitely many primes p. Claim. f has CM ⟹ ρ_{f} splits at p.

Proof. f has CM ⟹ there exists an imaginary quadratic field K/ℚ and a Hecke character ϕ on H = Gal(\overline{ℚ}/K) such that ρ_{f}|_{H} ∼ ( \begin{matrix}ϕ & 0 \\ 0 & ϕ̄ \end{matrix} ). Now p splits in K, so G_{p} ⊂ H. □

Guess: (??) f of CM type ⟺ ρ_{f} splits at p. What is known about the converse?

Converse in weight k = 2

Thm. (Serre) Say f ↔ elliptic curve E, p | N̸.

ρ_{f} splits at p ⟺ f has CM.

Def. Say p | N̸. Say ρ_{f} is totally split at p, if f is ordinary at ℘, and ρ_{f,℘} splits at p, for ALL primes ℘|p of K_{f}.

Prop. Say f ↔ abelian variety A and p | N̸.

ρ_{f} totally split at p ⟺ f has CM .

Def. Say p|N and the power of p dividing N and the conductor of χ are the same. Say f is totally split at p if for each prime ℘|p of K_{f}, f is ordinary at ℘ or ¯℘, and the corresponding local representation is split at p.

Prop. Say f ↔ A and p|N as above.

f totally split at p ⟺ f has CM .

The proofs are geometric (use p-divisible groups).

Converse in weights k > 2

What can one say in higher weights? Assume p > 2. Set M = ℚ(√p^{∗}) with p^{∗} = (−1)^{\frac{p−1}{2}} · p. Let N_{0} be an integer prime to p. Let S be the set of ℘-ordinary forms f of

weight k ≥ 2, level N = N_{0}p^{r}, r ≥ 0 satisfying two conditions:

1. ρ̄_{f} is absolutely irreducible on Gal(\overline{ℚ}/M),

2. f is p-distinguished ( ⟺ δ̄ ≠ ϵ̄ ). Main Thm. For all but finitely many f ∈ S

ρ_{f}|_{G_{p}} splits ⟺ f has CM.

Idea of Proof: Step 1: Partition S = ∪S_{F} by families F. Step 2: Each F either is CM or non-CM. Step 3: There is a Galois representation (Hida)

ρ_{F} : Gal(\overline{ℚ}/ℚ) → GL_{2}(K)

where K is a finite extension of the quotient field of Λ = ℤ_{p}[[X]], such that ρ_{F} specializes to ρ_{f} for f ∈ F, and ρ_{F} |_{G_{p}} is ‘upper triangular’:

ρ_{F} |_{G_{p}} ∼ (\begin{matrix}δ_{F} & u_{F} \\ 0 & ϵ_{F} \end{matrix} )

Theorem Λ. Assume ρ̄_{F} is absolutely irreducible on Gal(\overline{ℚ}/M) and F is p-distinguished.

ρ_{F} |_{G_{p}} splits ⟺ F is of CM type Step 4: Theorem Λ ⟹ Main Theorem.

Step 1. Λ-adic forms:

Recall p is an odd prime, N_{0} prime to p.

Fix Dirichlet characters:

• ψ mod N_{0}p,

• ω mod p cyclotomic character,

• χ_{ζ} : (ℤ/p^{r})^{×} → ℂ^{×} defined by noting

(ℤ/p^{r})^{×} = (ℤ/p)^{×} × ℤ/p^{r−1}

and setting

χ_{ζ} = 1 on (ℤ/p)^{×} and χ_{ζ}(1 + p) = ζ

where ζ is a primitive p^{r−1}-th root of 1, r ≥ 1.

• Finally, for k ≥ 1, set χ = ψω^{−k}χ_{ζ}. This is a character mod N_{0}p^{r}.

Let Λ = ℤ_{p}[[X]] = Iwasawa algebra.

Let L be the integral closure of Λ in a finite extension K of the quotient field of Λ.

We consider height one primes P : L → \overline{ℚ}_{p} which extend the homomorphism

Λ → \overline{ℚ}_{p}
X ↦ ζ(1 + p)^{k} − 1

where ζ is a p^{r−1}-th root of 1 and k ≥ 1.

Def. A Λ-adic cusp form, of level N_{0} and character ψ, is a formal q-expansion

F = ∞ ; ∑ ; n=1 a(n, F)q^{n} ∈ L[[q]]

such that for each homomorphism P : L → \overline{ℚ}_{p} as above, with k ≥ 2, the specialization

P (F) ∈ \overline{ℚ}_{p}[[q]]

is the q-expansion of a ℘-ordinary cusp form

f ∈ S_{k}(N, χ)

of level N = N_{0}p^{r} and character χ = ψω^{−k}χ_{ζ}.

Def. F is said to be primitive if it is normalized, an eigenform, and a newform.

Theorem. (Hida) Assume p > 2. 1.There are finitely many primitive Λ-adic forms F of level N_{0} (and some character ψ).

2. Each classical primitive ℘-ordinary from f of weight k ≥ 2 and level N_{0}p^{r} is the specialization of some F.

3. F in (2) is unique up to Galois conjugacy. Moral: We may write S = ⋃_{F} S_{F} where F varies through all Λ-adic forms F of level N_{0}. Step 2. CM vs. non-CM: Def/Prop. F has CM if one (all) specialization(s) f of weight k ≥ 2 are of CM type. Moral: The forms F are of two distinct types: CM or non-CM. Warning: A non-CM F may have weight 1 specializations of CM type.

Step 3. Greenberg for Λ-adic forms: Say p > 2. Let F be a Λ-adic form such that • F is p-distinguished, and,

• ρ̄_{F} is abs. irred. on Gal(\overline{ℚ}/M). Theorem Λ (expanded version). The following are equivalent:

1. ρ_{F} splits at p,

2. F has infinitely many classical weight 1 specializations,

3. F has infinitely many classical weight 1 specializations of CM type,

4. F has CM. Rmk. The theorem completely answers the Λadic analogue of Greenberg’s question (under the conditions above).

Proof of Theorem Λ:

(1) ⟹ (2): Say P : L → \overline{ℚ}_{p} is of weight 1, that is, it lies above X ↦ ζ(1 + p) − 1. Then

ρ_{P (F)}|_{I_{p}} ∼ (\begin{matrix}ψω^{−1}χ_{ζ} & 0 \\ 0 & 1 \end{matrix}) .

Theorem. (Buzzard, 2003) Let ρ : Gal(\overline{ℚ}/ℚ) → GL_{2}(O) be a continuous p-adic representation with

• ρ unramified outside a finite set of primes

• ρ̄ is modular

• ρ̄ is abs. irred. on Gal(\overline{ℚ}/M)

• ρ|_{G_{p}} ∼ (\begin{matrix}α & 0 \\ 0 & β \end{matrix}), α(I_{p}), β(I_{p}) finite, and ᾱ ≠ β̄.

Then: ρ ∼ ρ_{f}, for a classical weight 1 form f. Moral: F has ∞ly many weight 1 members!

(2) ⟹ (3): Wish to show that infinitely many of these weight 1 forms f are of CM type. Let ρ̃_{f} : Gal(\overline{ℚ}/ℚ) → PGL_{2}(ℂ). Then Image (ρ̃_{f}) is either

• cyclic,

• dihedral,

• A_{4}, S_{4}, or A_{5}. First case: Impossible by irreducibility. Last case: Possible only finitely many times.

Proof. ρ_{f}(I_{p}) has trivial intersection with the scalar matrices, so that ρ_{f}(I_{p}) injects into PGL_{2}(ℂ). But the groups A_{4}, S_{4} and A_{5} have bounded order.

Second case: Has to occur infinitely often! But now class field theory implies f can have real multiplication only finitely often, so f has complex multiplication infinitely often.

(3) ⟹ (4): Say F has infinitely many classical weight 1 specializations of CM type. Explicitly construct a Λ-adic form F′ of CM type which interpolates infinitely many of these specializations. Then F = F′ up to conjugacy, so F has CM.

(4) ⟹ (1): Easy! Theorem Λ is proved. □ Step 4: Recall S = ∪_{F} S_{F}. Case: F is of CM type: there is nothing to prove, since ρ_{f}|_{G_{p}} splits for all specializations f of weight ≥ 2. Case: F not of CM type: By Thm Λ, ρ_{F} is not split at p. It follows that ρ_{f} does not split at p for all but finitely many specializations f of weight at least 2. This prove the Main Theorem. □

Splitting mod p & Companion Forms

Think of f as a mod p form; assume 2 ≤ k ≤ p. When is ρ̄_{f} split at p?

Def. Say that a mod p eigenform g = ∑ b(n, g)q^{n} of weight k′ = p + 1 − k is a companion form for f if, for all n

n · a(n, f) = n^{k} · b(n, f) mod p.

Theorem. (Gross, Coleman-Voloch)

Say f is p-distinguished.

ρ̄_{f} splits at p ⟺ f has a companion g .

• If f is p-distinguished then ρ̄_{f} splits at p ⟺ ρ̄_{f} is tamely ramified at p.

• In fact C-V prove: ρ_{f} is tamely ramified at p ⟺ f has a companion form (except possibly in the case 2 = k = p).

Numerical examples: (Elkies)

Assume Image (ρ̄_{f}) contains SL_{2}(ℤ/p).

Form Companion p < 3500
Δ_{12}  none
Δ_{16}  397
Δ_{18}  271
Δ_{20}  139, 379
Δ_{22}  none
Δ_{26}  107

Can one lift this mod p information?

Claim. Assume f is p-distinguished and ρ̄_{f} is absolutely irreducible. Then

ρ̄_{f} not split at p ⟹ ρ_{f} not split at p.

Moral: Theorem Λ is really a theorem for the primes above. Even though there’s a mod p companion form, only finitely many of the members of the family to which Δ_{k} belongs can have locally split Galois representation!

Overconvergent forms

Let S_{k}(N_{0}) denote the overconvergent p-adic modular forms of tame level N_{0} and weight k ∈ ℤ.

S_{k}(N_{0}) is an ∞-dim’l p-adic Banach space, containing the classical forms of tame level N_{0}.

Let θ = q d/dq be the usual derivation.

f = ∑ a(n, f) q^{n} ↦ θ(f) = ∑ n · a(n, f) q^{n}.

It raises the weight by two and it is a result of Coleman that, for k ≥ 1:

θ^{k−1} : S_{2−k}(N_{0}) → S_{k}(N_{0}).

Def. The slope of f is v_{p}(a(p, f)).

So f is ordinary ⟺ f has slope 0. Note slope of θ^{k−1}g = slope of g + (k − 1).

Coleman’s question

Prop. (Coleman) Every classical CM eigenform f of weight k and slope k − 1 is of the form θ^{k−1}g where g is an ordinary eigenform of weight 2 − k

Question. (Coleman) Are there any non-CM classical eigenforms in the image of θ^{k−1}?

Prop. Let p ≥ 5. Let f be a ℘-ordinary form of weight k ≥ 2 and level N_{0}p^{r}. Assume

• f is p-distinguished,

• ρ̄_{f} is abs. irred. on Gal(\overline{ℚ}/M),

• a(q, f) = 0 for all q|N_{0}.

Then there is a form f′ of wt. k, slope k − 1:

f′ = θ^{k−1}g ⟺ ρ_{f} splits at p.

Corollary to Thm Λ. The answer to Coleman’s question is NO, for all except finitely many forms f′, as f varies through the weight ≥ 2 members of a non-CM family F as above.