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 Galois for low weight SMFs AFW, 21st March 2018 1 / 10
The classical case ∞ a n q n ∈ M k ( N , ǫ ) normalised Hecke eigenform, k ≥ 2 � f = n =0 Associated ℓ -adic Galois representation ρ ℓ : Gal( Q / Q ) → GL 2 ( Q ℓ ) unramified for all p ∤ ℓ N with det ρ ℓ = ǫχ k − 1 Tr ρ ℓ (Frob p ) = a p , ℓ Associated mod ℓ Galois representation ρ ℓ : Gal( Q / Q ) → GL 2 ( F ℓ ) Ariel Weiss (Sheffield) Irreducibility of Galois for low weight SMFs AFW, 21st March 2018 2 / 10
When are ρ ℓ and ρ ℓ irreducible? Example: a reducible ℓ -adic Galois representation ∞ 691 � σ 11 ( n ) q n ρ ℓ ∼ = 1 ⊕ χ 11 G 12 ( z ) = 65520 + ℓ n =1 Tr ρ ℓ (Frob p ) = 1 + p 11 a p = 1 + p 11 Theorem (Ribet) If f is cuspidal, then 1 ρ ℓ is irreducible for all ℓ ; 2 ρ ℓ is irreducible for all but finitely many ℓ . Example: a reducible mod ℓ Galois representation � ρ 691 ∼ τ ( n ) q n = 1 ⊕ χ 11 ∆( z ) = 1 + 691 n ≥ 2 Ariel Weiss (Sheffield) Irreducibility of Galois for low weight SMFs AFW, 21st March 2018 3 / 10
Genus 2 Siegel modular forms “Cuspidal automorphic representation of GSp 4 ( A Q ) + conditions at ∞ ” has weights ( k 1 , k 2 ), k 1 ≥ k 2 ≥ 2 has a level N has a character ǫ has Hecke operators T p and Hecke eigenvalues a p 4 types of cuspidal Siegel modular form: General Theta lifts/Automorphic inductions � Saito-Kurokawa/CAP reducible Galois representations Yoshida/endoscopic High weight: k 2 > 2 Low weight: k 2 = 2 Ariel Weiss (Sheffield) Irreducibility of Galois for low weight SMFs AFW, 21st March 2018 4 / 10
The high weight case: k 2 > 2 Associated ℓ -adic Galois representation ρ ℓ : Gal( Q / Q ) → GSp 4 ( Q ℓ ) unramified for all p ∤ ℓ N with sim ρ ℓ = ǫχ k 1 + k 2 − 3 Tr ρ ℓ (Frob p ) = a p , ℓ Associated mod ℓ Galois representation ρ ℓ : Gal( Q / Q ) → GSp 4 ( F ℓ ) ρ ℓ is always “kinda nice”, and is “nice” if ℓ ∤ N The Hecke eigenvalues satisfy the generalised Ramanujan conjecture Theorem 1 (Ramakrishnan) If ρ ℓ is nice and ℓ > 2( k 1 + k 2 − 3) + 1 , then ρ ℓ is irreducible; 2 (Dieulefait-Zenteno) ρ ℓ is irreducible for 100% of primes. Ariel Weiss (Sheffield) Irreducibility of Galois for low weight SMFs AFW, 21st March 2018 5 / 10
The low weight case: k 2 = 2 Associated ℓ -adic Galois representation ρ ℓ : Gal( Q / Q ) → GSp 4 ( Q ℓ ) unramified for all p ∤ ℓ N with sim ρ ℓ = ǫχ k 1 − 1 Tr ρ ℓ (Frob p ) = a p , ℓ Associated mod ℓ Galois representation ρ ℓ : Gal( Q / Q ) → GSp 4 ( F ℓ ) Theorem (W.) 1 If ρ ℓ is nice and ℓ > 2( k 1 − 1) + 1 , then ρ ℓ is irreducible; 2 ρ ℓ is irreducible for all but finitely many such primes. Theorem (W.) For 100% of primes ℓ , ρ ℓ is nice. Ariel Weiss (Sheffield) Irreducibility of Galois for low weight SMFs AFW, 21st March 2018 6 / 10
Irreducibility and modularity Sketch proof for modular forms. If f ∈ S k ( N , ǫ ) ↔ ρ ℓ and ρ ℓ is reducible then ⇒ ρ ℓ ≃ ψ ⊕ ϕχ k − 1 “kinda nice” = ℓ 1 CFT: ψ, ϕ correspond to Hecke (in this case Dirichlet) characters. 2 Write down another modular form ∞ ψ ( n E ψ,ϕ � � d ) ϕ ( d ) d k − 1 q n = a 0 + k n =1 d | n ) = ψ ( p ) + ϕ ( p ) p k − 1 = Tr ρ ℓ (Frob p ) = a p ( f ) . where a p ( E ψ,ϕ k Strong multiplicity one: f = E ψ,ϕ . k Idea: use the modularity of the subrepresentations of ρ ℓ to get a contradiction on the automorphic side. Ariel Weiss (Sheffield) Irreducibility of Galois for low weight SMFs AFW, 21st March 2018 7 / 10
Irreducibility and modularity II Idea: use the modularity of the subrepresentations of ρ ℓ to get a contradiction on the automorphic side. Key lemma (W.) Either ρ ℓ 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 ) → GL 2 ( Q ℓ ) is an irreducible, regular, odd and nice Galois representation, then ρ is potentially modular. If ρ ℓ is reducible then ρ ℓ ≃ σ 1 ⊕ σ 2 . If ρ ℓ is also nice, find automorphic representations π 1 , π 2 of GL 2 ( A K ) corresponding to σ 1 | K , σ 2 | K . Apply a standard L -functions argument. Ariel Weiss (Sheffield) Irreducibility of Galois for low weight SMFs AFW, 21st March 2018 8 / 10
Irreducibility in general Conjecture If π is a cuspidal automorphic representation of GL n ( A K ) then ρ ℓ 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 ρ ℓ is irreducible for 100% of primes. Ariel Weiss (Sheffield) Irreducibility of Galois for low weight SMFs AFW, 21st March 2018 9 / 10
Thank you for listening! Ariel Weiss (Sheffield) Irreducibility of Galois for low weight SMFs AFW, 21st March 2018 10 / 10
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