Modular mod ℓ Galois representations with exceptional Image (joint work with Ian Kiming)

Background: Classical examples

Congruences for coeffs of Δ(q)

Chebotarev method

Need to compute lots of a_{p}

Alternative method Start with a field K and find corresponding forms f

Outline of approach Reduce to a local problem Kiming ⇝ conductors and determinants Use structure of ρ|_{I_{ℓ}} to get weight

Some details of proof

Working for a particular K

Examples an S_{4} example: N = 1, k = 16, ℓ = 59 an A_{5} example: N = 23, k = 2, ℓ = 3 Question: For a prime ℓ > 2 and f a modular form (cuspidal new form, Hecke eigen form), is the modular Galois representation

\overline{ρ}_{f} : Gal(\overline{ℚ}/ℚ) → GL_{2}(𝔽_{ℓ^{n}})

exceptional?

i.e., is the image of the projectivization

˜\overline{ρ}_{f} : Gal(\overline{ℚ}/ℚ) → PGL_{2}(𝔽_{ℓ^{n}})

isomorphic to A_{4}, S_{4} or A_{5}?

Reminder:

\overline{ρ} is modular if

there is a modular form f = ∑ a_{n}q^{n} of weight k with Trace (\overline{ρ}(Frob _{p})) ≡ a_{p} mod ℓ det (\overline{ρ}(Frob _{p})) ≡ ε(p)p^{k−1} mod ℓ

Serre Invariant = (N, k, ε)

= (level, weight, det )

Classical Example: N = 1, k = 12, ℓ = 691

Δ = q ∞ ; ∏ ; n=1 (1 − q^{n})^{24} = ∞ ; ∑ ; n=1 τ(n)q^{n}

Ramanujan: τ(p) ≡ 1 + p^{11} mod 691 “Modern” Explanation: (Serre, Swinnerton-Dyer) The representation

\overline{ρ_{Δ}} : Gal(\overline{ℚ}/ℚ) → GL_{2}(𝔽_{691} )

is conjugate to

( \begin{matrix}χ^{11} & ∗ \\ & 1 \end{matrix} )

Where χ is the cycolotomic character.

Chebotarev:

For L/ℚ finite Galois,

C conjugacy class in Gal(L/K), # unramified primes p ≤ x with Frob _{p} ∈ C

= \frac{|C|}{|G|} · Li(x) + o(\frac{x}{log x}) .

E.g., f = (η(2z)η(4z))^{4}, weight 4, level 8, mod 11 Table counting values of (Trace (ρ(Frob _{p})), det (ρ(Frob _{p})) for p < 10000:

Determinant

trace 0 1 2 3 4 5 6 7 8 9 10
0  1 28 66 30 32 29 64 59 58 32 63
1 − 46 − 5 − − 27 − − − −
2 − 6 30 − 38 − − − − − −
3 − − − − − 6 − − − 40 30
4 − − − − 5 39 − − 31 − −
5 − − − 39 − − − 33 − 4 −
6 − − − 47 − − − 33 − 5 −
7 − − − − 4 43 − − 31 − −
8 − − − − − 5 − − − 37 30
9 − 5 28 − 42 − − − − − −
10 − 40 − 5 − − 33 − − − −

Problem: Need to compute many coefficients.

Alternative method (joint with Ian Kiming)

E.g., f mod ℓ form, level N ⇒ if \overline{ρ}_{f} is exceptional, image S_{4} then we have

\xymatrix{˜\overline{ρ} : Gal(\overline{ℚ}/ℚ) \ar[r] & Gal(K/ℚ) ≅ S_{4} \ar[r] & P GL_{2}(𝔽_{ℓ})}

K cut out by ker ˜ \overline{ρ} is unramified at p ∤ N, so only finitely many possibilities for K.

New Question: K/ℚ finite Galois, Gal(K/ℚ) ≅ to A_{4}, S_{4} or A_{5}: what are all possible Serre invariants (N, k, ε) of forms f mod ℓ with K = fixed field of ker (˜\overline{ρ})

Given such a list, is (N, k, ε) for f in the list? If yes, compute enough Trace (ρ(Frob _{p})) of representation from K to compare with coefficients of f.

Theorem (HAV+Kiming)

ρ̄ : G_{ℚ} ⟶ GL_{2}(\overline{𝔽}_{ℓ})

continuous, projective kernel field K/ℚ not totally real, Gal(K/ℚ) ≅ A_{4}, S_{4} or A_{5}. ℓ > 2.

Let S = primes≠ ℓ where K/ℚ ramifies.

Define n_{p}(ϕ_{p}) = n_{p}(K_{𝔭}/ℚ_{p}, ϕ_{p}) ∈ ℕ_{0}

for p ∈ S, and ϕ_{p} : (Units of ℤ_{p}) ⟶ ℂ^{×}, ϵ_{p} = ϵ(K_{𝔭}/ℚ_{p}) : (Units of ℤ_{p}) ⟶ \overline{𝔽}_{ℓ}^{×} , p ∈ S 𝔚_{ℓ} = 𝔚(K_{𝔩}/ℚ_{ℓ}) finite subset of ℕ .

Then: ρ̄ is mod ℓ modular with Serre invariant (N, k, ν), p|N ⇒⊆ S, and 2 ≤ k ≤ ℓ − 1

⇒ \{ \begin{matrix}k ∈ 𝔚_{ℓ}, \\ N = _{p∈S}^{∏} p^{n_{p}(ϕ_{|Ip})} \\ ν(r) = _{p∈S}^{∏} ϵ_{p}(r)^{−1} (ϕ_{|I_{p}}(r)^{−2} mod ℓ) \\ (for (r, N) = 1) \\ (for some ϕ : G_{ℚ} ⟶ ℂ^{×} of order prime to ℓ) \end{matrix}

Converse Assume Artin Conjecture. (N, k, ν) as above ⇒ ∃ a modular mod ℓ representation of G_{ℚ} with Serre invariant (N, k, ν).

Corollary For a modular Galois representation \overline{ρ} with projective kernel field K having Galois group A_{4}, S_{4} or A_{5}, Assuming Artin’s conjecture, we have

ℓ ≤ \{ \begin{matrix}3k − 2 & if G ≅ A_{4} \\ 4k − 3 & if G ≅ S_{4} \\ 5k − 4 & if G ≅ A_{5}. \end{matrix}

Outline of strategy for listing all (N, k, ε):

(0) Start with ˜ \overline{ρ} :

\xymatrix{Gal(\overline{ℚ}/ℚ) \ar[r]_{˜\overline{ρ}} & PGL_{2}(𝔽_{ℓ^{n}})}

(0) Start with ˜ \overline{ρ} .
(1) Construct a complex lift ρ̃ :

\xymatrix{ & PGL_{2}(ℂ) \ar[d] \\ Gal(\overline{ℚ}/ℚ) \ar[ur]_{ρ̃} \ar[r]_{˜\overline{ρ}} & PGL_{2}(𝔽_{ℓ^{n}})}

(0) Start with ˜ \overline{ρ} .
(1) Construct a complex lift ρ̃ .
(2) By theorem of Tate reduce to local problem:

\xymatrix{ & PGL_{2}(ℂ) \ar[d] \\ Gal(\overline{ℚ}_{p}/ℚ_{p}) \ar[ur]_{ρ̃_{p}} \ar[r]_{_{˜\overline{ρ}_{p}}} & PGL_{2}(𝔽_{ℓ^{n}})}

(0) Start with ˜ \overline{ρ} .
(1) Construct a complex lift ρ̃ .
(2) By theorem of Tate reduce to local problem.
(3) Give a lift ρ_{p,0} of ρ̃_{p} such that conductors and

determinants of all twists can be given (result of Kiming, after Weil, Buhler, Zink)

\xymatrix{ & GL_{2}(ℂ) \ar[r] & PGL_{2}(ℂ) \ar[d] \\ Gal(\overline{ℚ}_{p}/ℚ_{p}) \ar[ur]^{ρ_{p,0}} \ar[urr]_{ρ̃_{p}} \ar[rr]_{_{˜\overline{ρ}_{p}}} &  & PGL_{2}(𝔽_{ℓ^{n}})}

(0) Start with ˜ \overline{ρ} :
(1) Construct a complex lift ρ̃ :
(2) By theorem of Tate reduce to local problem:
(3) Give a lift ρ_{p,0} of ρ̃_{p} such that conductors and

determinants of all twists can be given (result of Kiming, after Weil, Buhler, Zink)

(4) Find what happens to conductors of \overline{ρ_{p,0}}  :

\xymatrix{ & GL_{2}(ℂ) \ar[r] \ar[d] & PGL_{2}(ℂ) \ar[d] \\ Gal(\overline{ℚ}_{p}/ℚ_{p}) \ar[r]^{\overline{ρ_{p,0}}} \ar[ur]^{ρ_{p,0}} \ar[rr]_{_{˜\overline{ρ}_{p}}} & GL_{2}(𝔽_{ℓ^{n}}) \ar[r] & PGL_{2}(𝔽_{ℓ^{n}})}

(0) Start with ˜ \overline{ρ} :
(1) Construct a complex lift ρ̃ :
(2) By theorem of Tate reduce to local problem:
(3) Give a lift ρ_{p,0} of ρ̃_{p} such that conductors and

determinants of all twists can be given (result of Kiming, after Weil, Buhler, Zink)

(4) Find what happens to conductors of \overline{ρ_{p,0}}  .
(5) Find possible weights (use form of \overline{ρ_{0}}|_{I_{ℓ}})

\xymatrix{ & GL_{2}(ℂ) \ar[r] \ar[d] & PGL_{2}(ℂ) \ar[d] \\ Gal(\overline{ℚ}_{ℓ}/ℚ_{ℓ}) \ar[r]^{\overline{ρ_{0}}} \ar[ur]^{ρ_{0}} \ar[rr]_{˜\overline{ρ}} & GL_{2}(𝔽_{ℓ^{n}}) \ar[r] & PGL_{2}(𝔽_{ℓ^{n}})}

(0) Start with ˜ \overline{ρ} :
(1) Construct a complex lift ρ̃ :
(2) By theorem of Tate reduce to local problem:
(3) Give a lift ρ_{p,0} of ρ̃_{p} such that conductors and

determinants of all twists can be given (result of Kiming, after Weil, Buhler, Zink)

(4) Find what happens to conductors of \overline{ρ_{p,0}}  .
(5) Find possible weights (use form of \overline{ρ_{0}}|_{I_{ℓ}}).
(6) Reverse direction: all such (N, k, ε) occur.

\xymatrix{ & GL_{2}(ℂ) \ar[d] \ar[r] & PGL_{2}(ℂ) \ar[d] \\ Gal(\overline{ℚ}/ℚ) \ar[r]^{\overline{ρ_{0}}} \ar[ur]^{ρ_{0}} \ar[rr]_{˜\overline{ρ}} & GL_{2}(𝔽_{ℓ^{n}}) \ar[r] & PGL_{2}(𝔽_{ℓ^{n}})}

Lifting representations For ℓ ∤ |Im˜\overline{ρ}|:

[figure: GL_{2}(ℂ); PGL_{2}(ℂ); Gal(\overline{ℚ}/ℚ); ^{^{(3)} ρ̃}; ^{(2)} ρ; (1); \overline{ρ}; _{_{(0)} ˜ \overline{ρ}}; GL_{2}(\overline{𝔽}_{ℓ}); PGL_{2}(\overline{𝔽}_{ℓ})]

(1) H^{2}(G_{ℚ}, \overline{𝔽}_{ℓ}^{×}) = 0

(2) Brauer Nesbitt applies when ℓ ∤ |Im˜\overline{ρ}| (3) The reduction ρ̃ of ρ is required lift Lifting representations For ℓ||Im˜\overline{ρ}|, E.g. Im˜\overline{ρ} ≅ A_{5} and ℓ = ℓ: Explicit description:

A_{5} = ⟨x, y | x^{2} = y^{5} = (xy)^{3} = 1⟩

x ↦ ( \begin{matrix}−c & ω \\ ω & c \end{matrix} ) , y ↦ ( \begin{matrix}ε^{2} & −ω \\ 0 & ε^{−2} \end{matrix} )

where

ε = \{ \begin{matrix}primitive 5th root of unity & ℓ ≠ 5 \\ 1 & ℓ = 5 \\ primitive 5th root of unity & for image in ℂ \end{matrix}

ω := ε + ε^{−1}, c := ε^{2} − ε^{−2}

For ℓ ≠ 5, there are two A_{5} representations to PGL_{2}(\overline{𝔽}_{ℓ}) (Dixon), switched by ε ↔ ε^{2}

The matrices define a complex representation and the reduced representation. (To obtain above matrices, take any complex representation, and then conjugate to make it integral. (E.g., Algorithm of Huppert))

Tate’s lifting theorem:

For a continuous representation

ρ̃: G_{ℚ} ⟶ PGL_{2}(ℂ)

Suppose we have lifts ρ_{p} of ρ̃_{p} ∀ primes p

\xymatrix{G_{ℚ_{p}} ≅ D_{p} \ar[r]^{ρ_{p}} \ar[dr]^{ρ̃_{p}} & GL_{2}(ℂ) \ar[d] \\  & PGL_{2}(ℂ)}

Then ∃! lift

ρ : G_{ℚ} ⟶ GL_{2}(ℂ)

with ρ_{|I_{p}} ≅ (ρ_{p})_{|I_{p}} ∀p

∗ ∗ ∗

Any lifts differ by a twist. If ρ_{p} is a lift of ρ̃_{p} so is ρ̃_{p} ⊗ ϕ, for any one dimensional character ϕ.

∗ ∗ ∗

Weil, Buhler, Zink, Kiming ⇝ description of ρ_{p,0} such that conductor and determinants of all possible twists of ρ_{p,0} can be given.

Local Lifting: example

Gal(K_{p}/ℚ_{p}) is cyclic, dihedral, A_{4} or S_{4}.

E.g.: Case (*): tamely ramified, dihedral, order 8, p ≠ 2

K_{p}_{⇝α:Gal(\overline{ℚ}_{p}/M)→ℂ^{×}}_{C_{4}} M_{C_{2}} _{Gal(M/ℚ_{p})≅⟨σ⟩} ℚ_{p}

Tate: H^{2}(G_{K}, ℚ_{p}/ℤ_{p}) = 0 ⇒ there is a lift.

Translation of problem via class field theory:

ρ̃|_{G_{M}} ∼ ( \begin{matrix}α \\ & 1 \end{matrix} )

want β : G_{M} → ℂ^{×}  want β̃ : M_{×} → ℂ_{×}
with ρ = Ind_{M}^{ℚ_{p}}(β)  ⇝
i.e. β_{σ}β_{−1} = α  β̃ (_{\frac{σx}{x}}) = α̃(x)
s.t. we can give  s.t. we can give
det (ρ ⊗ ϕ) β̃|_{U_{p}}
c_{p}(ρ ⊗ ϕ)  c_{p}(β̃ ∘ (ϕ̃ ∘ N_{K/Q_{p}}))

Degeneration of conductors: example

knowing conductor here

Gal(\overline{ℚ}_{p}/ℚ_{p}) ρ [figure]

\overline{ρ} [figure]

[figure]
GL_{2}(\overline{ℚ}_{ℓ})

[figure]

GL_{2}(\overline{𝔽}_{ℓ})

tells us the conductor here, by presevration of the Swan conductor

[figure] [figure]

[figure] [figure] [figure] [figure] [figure] [figure] [figure] [figure] [figure] [figure] [figure] [figure] [figure] [figure]

[figure]

c_{p}(ρ) − c_{p}(\overline{ρ}) = dim \overline{ρ}^{I_{p}} − dim ρ^{I_{p}}

Examine representations case by case, e.g., for case (*),

(ρ_{0,p} ⊗ ϕ_{p})_{|I_{p}} ∼ ( \begin{matrix}β·(ϕ_{p})_{|GM} & 0 \\ 0 & β^{σ}·(ϕ_{p})_{|GM} \end{matrix} ) ,

(ρ̄_{0,p} ⊗ ϕ̄_{p})_{|I_{p}} ∼ ( \begin{matrix}β̄·(ϕ̄_{p})_{|GM} & 0 \\ 0 & β̄^{σ}·(ϕ̄_{p})_{|GM} \end{matrix} )

So dim \overline{ρ}^{I_{p}} = dim ρ^{I_{p}} = 0 ⇒ c_{p}(ρ) − c_{p}(\overline{ρ})

The weight

Serre and Fontaine:

Ordinary case  Supersingular case
a_{ℓ} ≠ 0  a_{ℓ} = 0
\overline{ρ}_{ℓ}|_{I_{ℓ}} ∼ ( \begin{matrix}χ^{k−1} & ∗ \\ 0 & 1 \end{matrix} ) \overline{ρ}_{ℓ}|_{I_{ℓ}} ∼ ( \begin{matrix}(ψ′)^{k−1} & 0 \\ 0 & ψ^{k−1} \end{matrix} )

With ψ, ψ′ fundamental characters of level 2: ψ^{ℓ} = ψ′, ψ′^{ℓ} = ψ, order ℓ^{2} − 1, and ψψ′ = χ.

E.g., Case (*) ⇝ supersingular case. Order of ρ̃_{0,p}|_{I_{p}} = 4, so

ψ′^{k−1}ψ^{−(k−1)} = ψ^{(k−1)(ℓ−1)} has order 4

⇒ k − 1 = \frac{a(ℓ + 1)}{4} ( a odd )

twisting ⇒ k ≤ ℓ ⇒ a = 1 or 3 and

k = \frac{ℓ + 5}{4} , or k = \frac{3ℓ + 7}{4}

Reverse direction Need to show all possible weights occur, and the representations are modular. Artin conjecture ⇒ Modularity. all possible weights occur: theta cycles (supersingular case) or companion forms (ordinary case) (Coleman-Vollach)

Examples

k = 16, N = 1, ℓ = 59 (Harberland)

f(τ) = q (1 + 240 ∞ ; ∑ ; n=1 σ_{3}(n)) ∞ ; ∏ ; k=1 (1 − q^{k})^{24}

= q + 216q^{2} − 3348q^{3} + 13888q^{4} + ⋯

Numerical evidence ⇒ mod 59 representation is exceptional, projective image S_{4}. ∃! S_{4} extension K unramified outside 59, ∞, given by

x^{4} − x^{3} − 7x^{2} + 11x + 13.

Gal(K_{59} /ℚ_{59} ) ≅ D_{8} Above methods ⇒ N = 1 and k = 16 or 46. dim S_{16} (1) = 1, so f is exceptional. Example congruences:

a_{p} ≡ 0 mod 59 ⟺

x^{4} − x^{3} − 7x^{2} + 11x + 13 mod p has a quadratic factor Example 2 f = (η(2z)η(4z))^{4}, weight 4, level 8, mod 11 Expectation: this corresponds to an S_{4} extension of ℚ unramified outside 2 and 11 Take an S_{4} extension K (Found using class field theory to get S_{3} subextension, and Frey-Bayer to extend to S_{4}) given by

x^{4} − 2x^{3} − 4x^{2} + 16x − 13.

Gal(K_{2}/ℚ_{2}) ≅ S_{4}, and the extension K_{2} is ℚ_{2} (ζ_{3}, \sqrt[3]{2}, √1 + \sqrt[3]{2} , √1 + ζ_{3} \sqrt[3]{2} , √1 + ζ_{3}^{2} \sqrt[3]{2} ) And Gal(K_{11} /ℚ_{11} ) ≅ D_{4}.

Theorem ⇒ \{ \begin{matrix}c_{2}(ρ_{2,0} ⊗ ϕ_{2}) = max \{3, 2c_{2}(ϕ_{2})\} \\ k = 4 or 10 \end{matrix}

dim S_{4}(8) = 1 ⇒ f is exceptional.

Example 3

k = 2, N = 23, ℓ = 3

dim S_{2}(23) = 2 and it’s expected (Ribet) both forms are exceptional, image A_{5}, field given by

x^{5} + 3x^{3} + 6x^{2} + 9

Gal(K_{𝔭_{23}} /ℚ_{23} ) = D_{3} (unique) Gal(K_{𝔭_{3}}/ℚ_{3}) = D_{3} is totally ramified, peu ramifié (use tables in Buhler and Kiming-Basmaji)

Theorem ⇒ \{ \begin{matrix}c_{23} (ρ_{23,0} ⊗ ϕ_{23} ) = max \{1, 2c_{23} (ϕ_{23} )\} \\ ϵ_{23} = 1 \\ 𝔚_{3} = \{2\} \end{matrix}

Take ϕ_{23} = 1. There are two embeddings A_{5} ↪ PGL_{2}(ℂ), so we get both forms.