Monodromy for some rank two Galois representations over CM fields - - PowerPoint PPT Presentation

Monodromy for some rank two Galois representations over CM fields

Patrick Allen 1 James Newton 2

1University of Illinois at Urbana-Champaign 2Kings College London

Joint Math Mettings 2019

Galois representations attached to modular forms

Let f (z) =

n≥1 ane2πinz be a cuspidal newform of level N.

Theorem (Shimura, Deligne, Deligne–Serre)

For any prime ℓ and ι : Qℓ ∼ = C, there is a continuous semisimple Galois representation ρf ,ι : Gal(Q/Q) → GL2(Qℓ) such that for any p ∤ Nℓ, ρf ,ι is unramified at p and ι(tr ρf ,ι(Frobp)) = ap.

Question

What can we say about the restriction of ρf ,ι to Gal(Qp/Qp) when p|N?

The Weil group

Let K/Qp be finite with residue field k and let q = #k. Set GK = Gal(K/K), and let WK ⊆ GK be the Weil group of K: 1 IK GK Gal(k/k) ∼ = Z 1 1 IK WK {FrobZ

K}

1

Weil–Deligne representations

K as before. Let L be a characteristic 0 algebraically closed field. A Weil–Deligne representation of WK over L is a pair (r, N) with ◮ r : WK → GL(V ) ∼ = GLn(L) a representation of WK on a finite dimensional L-vector space V with ker(r|IK ) open. ◮ N ∈ End(V ), the monodromy operator, such that if Φ ∈ WK lifts the (geometric) Frobenius, then r(Φ)Nr(Φ)−1 = q−1N If (r, N) is a Weil–Deligne representation, it’s ◮ Frobenius semisimplification is (r, N)F-ss = (rss, N), ◮ semisimplification is (r, N)ss = (rss, 0).

Theorem (Grothendieck)

For ℓ = p, a continuous ρ : GK → GLn(Qℓ) 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 recK : Irr(GLn(K)) → WDrepn(WK) where ◮ Irr(GLn(K)) is the set of isomorphism classes of irreducible admissible representation of GLn(K) over C, ◮ WDrepn(WK) of isomorphism classes of n-dimensional Frobenius semisimple Weil–Deligne representations of WK

• ver C,

such that. . .

Example (n = 1)

For χ : K × → C×, recK(χ) = χ ◦ Art−1

K

where ArtK : K × ∼ − → W ab

K

is the Artin reciprocity map.

Local Langlands in rank 2

Identify χ : K × → C× with χ : WK → C× using ArtK. π = irreducible admissible representation of GL2(K). π recK(π) = (r, N) π(χ1, χ2) irreducible principal series, χi : K × → C× r = χ1 χ2

• , N = 0

χ ◦ det, χ : K × → C× r =

• χ|·|

1 2

χ|·|− 1

2

• , N = 0

St ⊗χ ◦ det χ : K × → C× r =

• χ|·|

1 2

χ|·|− 1

2

• , N =

1

• supercuspidal

r = irreducible, N = 0 0 → St ⊗χ ◦ det → n-IndGL2(K)

B

χ|·|

1 2 × χ|·|− 1 2 → χ ◦ det → 0

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 GLn(AF). Let ℓ be a prime and fix ι : Qℓ ∼ = C. Then there is a continuous semisimple representation ρπ,ι : GF → GLn(Qℓ) such that if p = ℓ is unramified for F and π and v|p in F, then WD(ρπ,ι|GFv )F-ss ⊗ι C ∼ = recFv (πv ⊗ |det|

1−n 2 ).

Conjecture (Local-global compatibility)

WD(ρπ,ι|GFv )F-ss ⊗ι C ∼ = recFv (πv ⊗ |det|

1−n 2 ) for all finite v in F.

Local-global compatibility

ρπ,ι : GF → GLn(Qℓ)

Conjecture (Local-global compatibility)

WD(ρπ,ι|GFv )F-ss ⊗ι C ∼ = recFv (πv ⊗ |det|

1−n 2 ) for all finite v in F.

Theorem (Harris–Taylor, Taylor–Yoshida, Shin, Barnet-Lamb–Gee–Geraghty–Taylor, Caraiani)

The conjecture is true if π is conjugate self-dual, i.e. πc ∼ = π∨.

Theorem (Varma)

Local-global compatibility holds for ρπ,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 GL2(AF) with F a CM field. There is a density

• ne set of primes ℓ such that for any ι : Qℓ ∼

= C, local global compatibility for ρπ,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 πv is a twist of

• Steinberg. Equivalently, that ρπ,ι|GFv has nontrivial unipotent

ramification. The ρπ,ι are constructed by using congruences from π × πc∨ to conjugate self dual cuspidal representations of GL2n(AF). 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 ρπ,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 πv is an unramified twist of St. By Varma, we just need to show that ρπ,ι|GFv is ramified. Assume otherwise. Using Taylor’s potential automorphy method and ALTs, we show that there is a finite extension F ′/F and cuspidal π1 on GLn(AF ′) unramified above v such that ρπ,ι|GF′ ∼ = ρπ1,ι mod ℓ. Using ALTs again, there is cuspidal π2 on GLn(AF ′) unramified above v such that ρπ,ι|GF′ ∼ = ρπ2,ι. This contradicts Varma.

Thank you