Irreducibility of Galois representations associated to low weight - - PowerPoint PPT Presentation
Irreducibility of Galois representations associated to low weight Siegel modular forms Ariel Weiss University of Sheffield a.weiss@sheffield.ac.uk 32nd Automorphic Forms Workshop 21st March 2018 Ariel Weiss (Sheffield) Irreducibility of
Ariel Weiss
University of Sheffield a.weiss@sheffield.ac.uk
32nd Automorphic Forms Workshop 21st March 2018
Ariel Weiss (Sheffield) Irreducibility of Galois for low weight SMFs AFW, 21st March 2018 1 / 10
f =
∞
anqn ∈ Mk(N, ǫ) normalised Hecke eigenform, k ≥ 2 Associated ℓ-adic Galois representation ρℓ : Gal(Q/Q) → GL2(Qℓ) unramified for all p ∤ ℓN with Tr ρℓ(Frobp) = ap, det ρℓ = ǫχk−1
ℓ
Associated mod ℓ Galois representation ρℓ : Gal(Q/Q) → GL2(Fℓ)
Ariel Weiss (Sheffield) Irreducibility of Galois for low weight SMFs AFW, 21st March 2018 2 / 10
Example: a reducible ℓ-adic Galois representation G12(z) = 691 65520 +
∞
σ11(n)qn ap = 1 + p11 ρℓ ∼ = 1 ⊕ χ11
ℓ
Tr ρℓ(Frobp) = 1 + p11
Theorem (Ribet)
If f is cuspidal, then
1 ρℓ is irreducible for all ℓ; 2 ρℓ is irreducible for all but finitely many ℓ.
Example: a reducible mod ℓ Galois representation ∆(z) = 1 +
τ(n)qn ρ691 ∼ = 1 ⊕ χ11
691
Ariel Weiss (Sheffield) Irreducibility of Galois for low weight SMFs AFW, 21st March 2018 3 / 10
“Cuspidal automorphic representation of GSp4(AQ) + conditions at ∞” has weights (k1, k2), k1 ≥ k2 ≥ 2 has a level N has a character ǫ has Hecke operators Tp and Hecke eigenvalues ap 4 types of cuspidal Siegel modular form: General Theta lifts/Automorphic inductions Saito-Kurokawa/CAP Yoshida/endoscopic
High weight: k2 > 2 Low weight: k2 = 2
Ariel Weiss (Sheffield) Irreducibility of Galois for low weight SMFs AFW, 21st March 2018 4 / 10
Associated ℓ-adic Galois representation ρℓ : Gal(Q/Q) → GSp4(Qℓ) unramified for all p ∤ ℓN with Tr ρℓ(Frobp) = ap, simρℓ = ǫχk1+k2−3
ℓ
Associated mod ℓ Galois representation ρℓ : Gal(Q/Q) → GSp4(Fℓ) ρℓ is always “kinda nice”, and is “nice” if ℓ ∤ N The Hecke eigenvalues satisfy the generalised Ramanujan conjecture
Theorem
1 (Ramakrishnan) If ρℓ is nice and ℓ > 2(k1 + k2 − 3) + 1, then ρℓ is
irreducible;
2 (Dieulefait-Zenteno) ρℓ is irreducible for 100% of primes. Ariel Weiss (Sheffield) Irreducibility of Galois for low weight SMFs AFW, 21st March 2018 5 / 10
Associated ℓ-adic Galois representation ρℓ : Gal(Q/Q) → GSp4(Qℓ) unramified for all p ∤ ℓN with Tr ρℓ(Frobp) = ap, sim ρℓ = ǫχk1−1
ℓ
Associated mod ℓ Galois representation ρℓ : Gal(Q/Q) → GSp4(Fℓ)
Theorem (W.)
1 If ρℓ is nice and ℓ > 2(k1 − 1) + 1, then ρℓ is irreducible; 2 ρℓ is irreducible for all but finitely many such primes.
Theorem (W.)
For 100% of primes ℓ, ρℓ is nice.
Ariel Weiss (Sheffield) Irreducibility of Galois for low weight SMFs AFW, 21st March 2018 6 / 10
Sketch proof for modular forms.
If f ∈ Sk(N, ǫ) ↔ ρℓ and ρℓ is reducible then “kinda nice” = ⇒ ρℓ ≃ ψ ⊕ ϕχk−1
ℓ
1 CFT: ψ, ϕ correspond to Hecke (in this case Dirichlet) characters. 2 Write down another modular form
E ψ,ϕ
k
= a0 +
∞
d|n
ψ(n d )ϕ(d)dk−1 qn where ap(E ψ,ϕ
k
) = ψ(p) + ϕ(p)pk−1 = Tr ρℓ(Frobp) = ap(f ). Strong multiplicity one: f = E ψ,ϕ
k
. Idea: use the modularity of the subrepresentations of ρℓ to get a contradiction on the automorphic side.
Ariel Weiss (Sheffield) Irreducibility of Galois for low weight SMFs AFW, 21st March 2018 7 / 10
Idea: use the modularity of the subrepresentations of ρℓ to get a contradiction on the automorphic side.
Key lemma (W.)
Either ρℓ is irreducible, or it splits as a direct sum of two-dimensional representations which are irreducible, regular and odd.
Theorem (Taylor)
If ℓ is sufficiently large, and ρ : Gal(Q/Q) → GL2(Qℓ) is an irreducible, regular, odd and nice Galois representation, then ρ is potentially modular. If ρℓ is reducible then ρℓ ≃ σ1 ⊕ σ2. If ρℓ is also nice, find automorphic representations π1, π2 of GL2(AK) corresponding to σ1|K, σ2|K. Apply a standard L-functions argument.
Ariel Weiss (Sheffield) Irreducibility of Galois for low weight SMFs AFW, 21st March 2018 8 / 10
Conjecture
If π is a cuspidal automorphic representation of GLn(AK) then ρℓ is irreducible for all primes. Known results: n = 2: Ribet n = 3: Blasius-Rogawski if K totally real, π essentially self dual Partial results: (Barnet-Lamb–Gee–Geraghty–Taylor) if K is CM and π is “extremely regular”, then ρℓ is irreducible for 100% of primes.
Ariel Weiss (Sheffield) Irreducibility of Galois for low weight SMFs AFW, 21st March 2018 9 / 10
Ariel Weiss (Sheffield) Irreducibility of Galois for low weight SMFs AFW, 21st March 2018 10 / 10