Monodromy for some rank two Galois representations over CM fields Patrick Allen 1 James Newton 2 1 University of Illinois at Urbana-Champaign 2 Kings College London Joint Math Mettings 2019
Galois representations attached to modular forms Let f ( z ) = � n ≥ 1 a n e 2 π inz be a cuspidal newform of level N . Theorem (Shimura, Deligne, Deligne–Serre) For any prime ℓ and ι : Q ℓ ∼ = C , there is a continuous semisimple Galois representation ρ f ,ι : Gal ( Q / Q ) → GL 2 ( Q ℓ ) such that for any p ∤ N ℓ , ρ f ,ι is unramified at p and ι (tr ρ f ,ι ( Frob p )) = a p . Question What can we say about the restriction of ρ f ,ι to Gal ( Q p / Q p ) when p | N?
The Weil group Let K / Q p be finite with residue field k and let q = # k . Set G K = Gal ( K / K ), and let W K ⊆ G K be the Weil group of K : Gal ( k / k ) ∼ = � 1 I K G K Z 1 { Frob Z 1 K } 1 I K W K
Weil–Deligne representations K as before. Let L be a characteristic 0 algebraically closed field. A Weil–Deligne representation of W K over L is a pair ( r , N ) with ◮ r : W K → GL( V ) ∼ = GL n ( L ) a representation of W K on a finite dimensional L -vector space V with ker( r | I K ) open. ◮ N ∈ End( V ), the monodromy operator , such that if Φ ∈ W K lifts the (geometric) Frobenius, then r (Φ) Nr (Φ) − 1 = q − 1 N If ( r , N ) is a Weil–Deligne representation, it’s ◮ Frobenius semisimplification is ( r , N ) F - ss = ( r ss , N ), ◮ semisimplification is ( r , N ) ss = ( r ss , 0). Theorem (Grothendieck) For ℓ � = p, a continuous ρ : G K → GL n ( Q ℓ ) naturally determines a Weil–Deligne representation WD( ρ ) that determines ρ up to isomorphism.
The local Langlands correspondence Theorem (Harris–Taylor, Henniart) There is a unique bijection rec K : Irr(GL n ( K )) → WDrep n ( W K ) where ◮ Irr(GL n ( K )) is the set of isomorphism classes of irreducible admissible representation of GL n ( K ) over C , ◮ WDrep n ( W K ) of isomorphism classes of n-dimensional Frobenius semisimple Weil–Deligne representations of W K over C , such that. . . Example ( n = 1) For χ : K × → C × , rec K ( χ ) = χ ◦ Art − 1 where Art K : K × ∼ → W ab − K K is the Artin reciprocity map.
Local Langlands in rank 2 Identify χ : K × → C × with χ : W K → C × using Art K . π = irreducible admissible representation of GL 2 ( K ). π rec K ( π ) = ( r , N ) � χ 1 � π ( χ 1 , χ 2 ) irreducible principal series, r = , N = 0 χ 2 χ i : K × → C × � � 1 χ ◦ det, χ |·| 2 r = , N = 0 χ : K × → C × χ |·| − 1 2 � � � 0 � 1 St ⊗ χ ◦ det χ |·| 1 2 r = , N = χ : K × → C × χ |·| − 1 0 0 2 supercuspidal r = irreducible, N = 0 1 2 × χ |·| − 1 0 → St ⊗ χ ◦ det → n-Ind GL 2 ( K ) 2 → χ ◦ det → 0 χ |·| B
Local-global compatibility √ Let F be a CM number field (e.g. Q ( − D )). Theorem (Harris–Lan–Taylor–Thorne) Let π be a regular algebraic cuspidal automorphic representation of GL n ( A F ) . Let ℓ be a prime and fix ι : Q ℓ ∼ = C . Then there is a continuous semisimple representation ρ π,ι : G F → GL n ( Q ℓ ) such that if p � = ℓ is unramified for F and π and v | p in F, then WD( ρ π,ι | G Fv ) F - ss ⊗ ι C ∼ 1 − n 2 ) . = rec F v ( π v ⊗ | det | Conjecture (Local-global compatibility) WD( ρ π,ι | G Fv ) F - ss ⊗ ι C ∼ 1 − n 2 ) for all finite v in F. = rec F v ( π v ⊗ | det |
Local-global compatibility ρ π,ι : G F → GL n ( Q ℓ ) Conjecture (Local-global compatibility) WD( ρ π,ι | G Fv ) F - ss ⊗ ι C ∼ 1 − n 2 ) for all finite v in F. = rec F v ( π v ⊗ | det | Theorem (Harris–Taylor, Taylor–Yoshida, Shin, Barnet-Lamb–Gee–Geraghty–Taylor, Caraiani) The conjecture is true if π is conjugate self-dual, i.e. π c ∼ = π ∨ . Theorem (Varma) Local-global compatibility holds for ρ π,ι at all v ∤ ℓ up to the monodromy operator N. Question What about N in the non-conjugate self-dual case?
Monodromy in rank 2 Theorem (A.–Newton) Let π be a regular algebraic weight 0 cuspidal automorphic representation of GL 2 ( A F ) with F a CM field. There is a density one set of primes ℓ such that for any ι : Q ℓ ∼ = C , local global compatibility for ρ π,ι holds at all v ∤ ℓ . Remarks 1. “regular algebraic weight 0 ” is the analogue of classical modular forms of weight 2 . 2. In principal, should be able to improve “density one” to “all but finitely many.”
What’s the difficulty? Goal: Need to show N � = 0 at v for which π v is a twist of Steinberg. Equivalently, that ρ π,ι | G Fv has nontrivial unipotent ramification. The ρ π,ι are constructed by using congruences from π × π c ∨ to conjugate self dual cuspidal representations of GL 2 n ( A F ). Can keep track of characteristic polynomials with this method, so know things up to semisimplification. Proof that N � = 0 in the conjugate self-dual case relies crucially on the fact that we can find ρ π,ι in the ´ etale cohomology of a Shimura variety. This is not possible in the non-conjugate self dual case.
N � = 0 via automorphy lifting theorems We take our inspiration by an approach of Luu using automorphy lifting theorems (ALTs). Main ingredient: ALTs of A.–Calegari–Caraiani–Gee–Helm–Le Hung–Newton–Scholze–Taylor–Thorne. Can assume π v is an unramified twist of St. By Varma, we just need to show that ρ π,ι | G Fv is ramified. Assume otherwise. Using Taylor’s potential automorphy method and ALTs, we show that there is a finite extension F ′ / F and cuspidal π 1 on GL n ( A F ′ ) unramified above v such that ρ π,ι | G F ′ ∼ = ρ π 1 ,ι mod ℓ . Using ALTs again, there is cuspidal π 2 on GL n ( A F ′ ) unramified above v such that ρ π,ι | G F ′ ∼ = ρ π 2 ,ι . This contradicts Varma.
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